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arxiv: 2605.23862 · v2 · pith:IM2PJPZ2new · submitted 2026-05-22 · 🪐 quant-ph · gr-qc

Indefinite probabilities in quantum spacetime: A deepening of unpredictability

Vittorio D'Esposito , Giuseppe Fabiano , Domenico Frattulillo This is my paper

Pith reviewed 2026-06-30 15:51 UTC · model grok-4.3

classification 🪐 quant-ph gr-qc
keywords indefinite probabilitiesquantum groupsSU_q(2)non-commutative spacetimespin measurementsStern-Gerlach apparatusquantum gravityuncertainty principle
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The pith

Using the SU_q(2) quantum group to describe rotational symmetry turns spin measurement probabilities into non-commuting operators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to show that describing rotational symmetry for spin-1/2 systems via the SU_q(2) quantum group makes the probabilities of measurement outcomes into non-commuting operators. This produces an uncertainty principle between those operators and realizes indefinite probabilities. The same non-commutativity appears in the entries of the rotation matrix that relates two observers' reference frames. A sympathetic reader would care because the effect prevents sharp determination of relative orientation and ties the indefiniteness directly to the symmetry structure rather than to the quantum state alone.

Core claim

Employing the SU_q(2) quantum group to describe rotational symmetry for spin-1/2 systems and Stern-Gerlach apparatuses leads to the description of probabilities of outcomes of spin measurements in terms of non-commuting operators, realizing a notion of indefinite probabilities reflected in the non-commutativity of the entries of the rotation matrix relating the reference frames of two observers.

What carries the argument

The SU_q(2) quantum group as the deformed description of rotational symmetry, which renders the probability operators non-commuting and likewise renders the rotation-matrix entries non-commuting.

If this is right

  • An uncertainty principle holds between different probability operators.
  • The rotation matrix relating the reference frames of two observers has non-commuting entries.
  • Observers cannot sharply measure their relative orientation.
  • Indefiniteness of probabilities is tied to the quantum-group structure of spacetime symmetries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that classical notions of fixed relative orientation lose meaning once quantum-gravity corrections to symmetry are included.
  • Similar non-commutativity of probabilities might appear in other measurement contexts that involve deformed symmetries.
  • Precision Stern-Gerlach experiments at energies where quantum-gravity effects on rotation become detectable could in principle reveal the non-commutativity.
  • The mechanism offers one concrete way indefinite probabilities could arise without invoking additional hidden variables.

Load-bearing premise

The assumption that the SU_q(2) quantum group supplies the correct description of rotational symmetry for spin-1/2 systems in the low-energy limit of quantum gravity.

What would settle it

A direct computation or measurement showing that the probability operators for spin outcomes commute when rotational symmetry is modeled by SU_q(2) in a quantum-spacetime setting.

Figures

Figures reproduced from arXiv: 2605.23862 by Domenico Frattulillo, Giuseppe Fabiano, Vittorio D'Esposito.

Figure 1
Figure 1. Figure 1: FIG. 1. A pictorial representation of the effect of probability measurements on spacetime itself. On the left-hand side, the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
read the original abstract

Non-commutative spacetime and quantum groups have been argued to capture non-classical features of spacetime and its symmetries in the low-energy limit of quantum gravity. In this letter, we show that employing the $SU_q(2)$ quantum group to describe rotational symmetry for spin-$\frac{1}{2}$ systems and Stern-Gerlach apparatuses leads to the description of probabilities of outcomes of spin measurements in terms of non-commuting operators. As a result, we obtain an uncertainty principle between different probability operators, realizing a notion of indefinite probabilities. This is then reflected in the non-commutativity of the entries of the rotation matrix relating the reference frames of two observers, hence fundamentally preventing them from sharply measuring their relative orientation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper claims that employing the SU_q(2) quantum group to describe rotational symmetry for spin-1/2 systems and Stern-Gerlach apparatuses leads to probabilities of spin measurement outcomes being expressed via non-commuting operators. This yields an uncertainty principle between probability operators (indefinite probabilities) that manifests as non-commutativity in the entries of the rotation matrix relating two observers' frames, thereby preventing sharp determination of their relative orientation.

Significance. If the central derivation holds and the foundational premise is justified, the result would extend quantum-group techniques to produce a concrete operator-level notion of indefinite probabilities, offering a new angle on unpredictability in quantum spacetime. The work is positioned as a letter, so its significance depends on whether the non-commutativity follows rigorously from the q-deformed algebra rather than from the choice of symmetry group alone.

major comments (2)
  1. [Abstract] Abstract (opening sentence): the claim that SU_q(2) supplies the appropriate description of rotational symmetry for spin-1/2 systems in the low-energy limit of quantum gravity is invoked without derivation or explicit link to any concrete quantum-gravity construction (e.g., a limit of loop quantum gravity, causal dynamical triangulations, or a specific non-commutative geometry model). This premise is load-bearing for the physical interpretation of the subsequent non-commuting probability operators.
  2. [Abstract] Abstract (final paragraph): the transition from non-commuting probability operators to the non-commutativity of the rotation-matrix entries is presented as a direct consequence, but without the explicit operator algebra or commutation relations shown, it is impossible to verify whether the non-commutativity is an artifact of the framing or follows from the SU_q(2) coproduct and representation theory.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough reading and valuable comments, which help clarify the presentation of our results. We address each major comment below and indicate where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract (opening sentence): the claim that SU_q(2) supplies the appropriate description of rotational symmetry for spin-1/2 systems in the low-energy limit of quantum gravity is invoked without derivation or explicit link to any concrete quantum-gravity construction (e.g., a limit of loop quantum gravity, causal dynamical triangulations, or a specific non-commutative geometry model). This premise is load-bearing for the physical interpretation of the subsequent non-commuting probability operators.

    Authors: We acknowledge that the opening sentence states the premise without an explicit derivation from a specific quantum-gravity model. The choice of SU_q(2) is grounded in the existing literature on quantum groups as effective symmetries in the low-energy regime (e.g., connections to non-commutative geometries and deformed algebras in loop quantum gravity approaches). To address this, we will revise the abstract and add a short paragraph with targeted references in the introduction, making the foundational motivation more explicit while preserving the letter format. revision: yes

  2. Referee: [Abstract] Abstract (final paragraph): the transition from non-commuting probability operators to the non-commutativity of the rotation-matrix entries is presented as a direct consequence, but without the explicit operator algebra or commutation relations shown, it is impossible to verify whether the non-commutativity is an artifact of the framing or follows from the SU_q(2) coproduct and representation theory.

    Authors: The full manuscript derives this transition explicitly in Sections 3–4 from the non-cocommutative coproduct of SU_q(2) and the associated representation theory; the probability operators are constructed via the deformed generators, and the rotation-matrix entries inherit non-commutativity directly from [Δ(J_i), Δ(J_j)] relations. The abstract is a summary, but we agree it can better signpost the origin. We will revise the abstract to include a brief pointer to the key coproduct non-cocommutativity and ensure the derivation is highlighted more clearly in the main text. revision: partial

Circularity Check

0 steps flagged

No circularity; result follows from external premise of SU_q(2) applicability without internal reduction.

full rationale

The abstract states that non-commutative spacetime and quantum groups 'have been argued to capture' features in low-energy quantum gravity, then shows consequences of employing SU_q(2) for spin-1/2 systems. No equations are present, no self-citations are quoted as load-bearing, and no step reduces the derived non-commuting probabilities or rotation-matrix non-commutativity to a fitted input or self-defined premise by construction. The derivation is therefore independent of its own outputs given the stated external premise.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that SU_q(2) correctly captures rotational symmetry in this quantum-spacetime setting; no free parameters or invented entities are visible from the abstract alone.

axioms (1)
  • domain assumption The SU_q(2) quantum group describes rotational symmetry for spin-1/2 systems and Stern-Gerlach apparatuses in the low-energy limit of quantum gravity.
    Invoked in the first sentence of the abstract as the modeling choice that produces the claimed non-commutativity.
invented entities (1)
  • indefinite probabilities no independent evidence
    purpose: To name the situation in which spin-measurement probabilities are represented by non-commuting operators.
    Introduced as the direct consequence of the non-commutativity; no independent evidence supplied in the abstract.

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Reference graph

Works this paper leans on

48 extracted references · 39 canonical work pages · 21 internal anchors

  1. [1]

    From this perspective, the incompatibility of proba- bility measurements associated with non-commuting di- rections reflects the more fundamental fact that the ge- ometry on which these probabilities are defined is itself quantum, and is therefore unavoidably disturbed by the measurement process. A. Communication protocol between Alice and Bob Now we retu...

    2024

  2. [2]

    Reference frames, superselection rules, and quantum information

    S.D. Bartlett, T. Rudolph and R.W. Spekkens, Reference frames, superselection rules, and quantum information,Rev. Mod. Phys.79(2007) 555 [quant-ph/0610030]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  3. [3]

    Amelino-Camelia, V

    G. Amelino-Camelia, V. D’Esposito, G. Fabiano, D. Frattulillo, P.A. H¨ ohn and F. Mercati,Quantum Euler angles and agency-dependent space-time,PTEP 2024(2024) 033A01 [2211.11347]

    work page arXiv 2024

  4. [4]

    D’Esposito, G

    V. D’Esposito, G. Fabiano, D. Frattulillo and F. Mercati,Doubly Quantum Mechanics,Quantum9 (2025) 1721 [2412.05997]

    work page arXiv 2025

  5. [5]

    Discreteness of area and volume in quantum gravity

    C. Rovelli and L. Smolin,Discreteness of area and volume in quantum gravity,Nucl. Phys. B442(1995) 593 [gr-qc/9411005]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  6. [6]

    Quantum Theory of Gravity I: Area Operators

    A. Ashtekar and J. Lewandowski,Quantum theory of geometry. 1: Area operators,Class. Quant. Grav.14 (1997) A55 [gr-qc/9602046]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  7. [7]

    Spectra of Length and Area in 2+1 Lorentzian Loop Quantum Gravity

    L. Freidel, E.R. Livine and C. Rovelli,Spectra of length and area in (2+1) Lorentzian loop quantum gravity, Class. Quant. Grav.20(2003) 1463 [gr-qc/0212077]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  8. [8]

    Discreteness of area in noncommutative space

    G. Amelino-Camelia, G. Gubitosi and F. Mercati, Discreteness of area in noncommutative space,Phys. Lett. B676(2009) 180 [0812.3663]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  9. [9]

    Algebraic approach to quantum gravity II: noncommutative spacetime

    S. Majid,Algebraic approach to quantum gravity. II: Noncommutative spacetime, [hep-th/0604130]

    work page internal anchor Pith review Pith/arXiv arXiv

  10. [10]

    Localization and Reference Frames in $\kappa$-Minkowski Spacetime

    F. Lizzi, M. Manfredonia, F. Mercati and T. Poulain, Localization and Reference Frames inκ-Minkowski Spacetime,Phys. Rev. D99(2019) 085003 [1811.08409]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  11. [11]

    Algebraic approach to quantum gravity IV: applications

    S. Majid,Algebraic approach to quantum gravity IV: applications, [2604.06118]

    work page internal anchor Pith review Pith/arXiv arXiv

  12. [12]

    Snyder,Quantized space-time,Phys

    H.S. Snyder,Quantized space-time,Phys. Rev.71 (1947) 38

    1947

  13. [13]

    String Theory and Noncommutative Geometry

    N. Seiberg and E. Witten,String theory and noncommutative geometry,JHEP09(1999) 032 [hep-th/9908142]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  14. [14]

    Noncommutative Field Theory

    M.R. Douglas and N.A. Nekrasov,Noncommutative field theory,Rev. Mod. Phys.73(2001) 977 [hep-th/0106048]

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  15. [15]

    Noncommutative Spacetime, Stringy Spacetime Uncertainty Principle, and Density Fluctuations

    R. Brandenberger and P.-M. Ho,Noncommutative space-time, stringy space-time uncertainty principle, and density fluctuations,Phys. Rev. D66(2002) 023517 [hep-th/0203119]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  16. [16]

    3d Quantum Gravity and Effective Non-Commutative Quantum Field Theory

    L. Freidel and E.R. Livine,3D Quantum Gravity and Effective Noncommutative Quantum Field Theory, Phys. Rev. Lett.96(2006) 221301 [hep-th/0512113]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  17. [17]

    Emergent non-commutative matter fields from Group Field Theory models of quantum spacetime

    D. Oriti,Emergent non-commutative matter fields from Group Field Theory models of quantum spacetime,J. Phys. Conf. Ser.174(2009) 012047 [0903.3970]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  18. [18]

    Spacetime-noncommutativity regime of Loop Quantum Gravity

    G. Amelino-Camelia, M.M. da Silva, M. Ronco, L. Cesarini and O.M. Lecian, Spacetime-noncommutativity regime of Loop Quantum Gravity,Phys. Rev. D95(2017) 024028 [1605.00497]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  19. [19]

    Quantum Field Theory on Noncommutative Spaces

    R.J. Szabo,Quantum field theory on noncommutative spaces,Phys. Rept.378(2003) 207 [hep-th/0109162]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  20. [20]

    Quantum Spacetime Phenomenology

    G. Amelino-Camelia,Quantum-Spacetime Phenomenology,Living Rev. Rel.16(2013) 5 [0806.0339]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  21. [21]

    Majid,Foundations of Quantum Group Theory, Cambridge University Press (1995), 10.1017/CBO9780511613104

    S. Majid,Foundations of Quantum Group Theory, Cambridge University Press (1995), 10.1017/CBO9780511613104

    work page doi:10.1017/cbo9780511613104 1995

  22. [22]

    Arzano, V

    M. Arzano, V. D’Esposito and G. Gubitosi, Fundamental decoherence from quantum spacetime, Commun. Phys.6(2023) 242 [2208.14119]. 8

    work page arXiv 2023

  23. [23]

    Ballesteros, D

    A. Ballesteros, D. Fernandez-Silvestre, F. Giacomini and G. Gubitosi,Quantum Galilei group as quantum reference frame transformations, [2504.00569]

    work page arXiv

  24. [24]

    Gubitosi, F

    G. Gubitosi, F. Lizzi, J.J. Relancio and P. Vitale, Double quantization,Phys. Rev. D105(2022) 126013 [2112.11401]

    work page arXiv 2022

  25. [25]

    Arzano, A

    M. Arzano, A. Del Prete and D. Frattulillo,Quantum Evolution of Hopf Algebra Hamiltonians, [2602.07887]

    work page arXiv

  26. [26]

    Arzano, G

    M. Arzano, G. Chirco and J. Kowalski-Glikman,Bias in Local Spin Measurements from Deformed Symmetries, [2603.08618]

    work page arXiv

  27. [27]

    Quantum mechanics and the covariance of physical laws in quantum reference frames

    F. Giacomini, E. Castro-Ruiz and v. Brukner,Quantum mechanics and the covariance of physical laws in quantum reference frames,Nature Commun.10(2019) 494 [1712.07207]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  28. [28]

    Woronowicz,Compact matrix pseudogroups, Commun

    S.L. Woronowicz,Compact matrix pseudogroups, Commun. Math. Phys.111(1987) 613

    1987

  29. [29]

    Quantum deformation of quantum gravity

    S. Major and L. Smolin,Quantum deformation of quantum gravity,Nucl. Phys. B473(1996) 267 [gr-qc/9512020]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  30. [30]

    Spin Foam Models and the Classical Action Principle

    L. Freidel and K. Krasnov,Spin foam models and the classical action principle,Adv. Theor. Math. Phys.2 (1999) 1183 [hep-th/9807092]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  31. [31]

    Girelli and M

    F. Girelli and M. Laudonio,Group field theory on quantum groups, [2205.13312]

    work page arXiv

  32. [32]

    A note on the geometrical interpretation of quantum groups and non-commutative spaces in gravity

    E. Bianchi and C. Rovelli,Note on the geometrical interpretation of quantum groups and non-commutative spaces in gravity,Phys. Rev. D84(2011) 027502 [1105.1898]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  33. [33]

    Hoehn, A.R.H

    P.A. H¨ ohn, A.R. Smith and M.P. Lock,Trinity of relational quantum dynamics,Phys. Rev. D104(2021) 066001 [1912.00033]

    work page arXiv 2021

  34. [34]

    Giacomini, E

    F. Giacomini, E. Castro-Ruiz and ˇC. Brukner, Relativistic quantum reference frames: the operational meaning of spin,Phys. Rev. Lett.123(2019) 090404 [1811.08228]

    work page arXiv 2019

  35. [35]

    Streiter, F

    L.F. Streiter, F. Giacomini and ˇC. Brukner,Relativistic Bell Test within Quantum Reference Frames,Phys. Rev. Lett.126(2021) 230403 [2008.03317]

    work page arXiv 2021

  36. [36]

    de la Hamette and T

    A.-C. de la Hamette and T.D. Galley,Quantum reference frames for general symmetry groups,Quantum 4(2020) 367 [2004.14292]

    work page arXiv 2020

  37. [37]

    Mikusch, L

    M. Mikusch, L.C. Barbado and v. Brukner, Transformation of spin in quantum reference frames, Phys. Rev. Res.3(2021) 043138 [2103.05022]

    work page arXiv 2021

  38. [38]

    Ahmad Ali, T

    S. Ahmad Ali, T.D. Galley, P.A. H¨ ohn, M.P.E. Lock and A.R.H. Smith,Quantum relativity of subsystems, Phys. Rev. Lett.128(2022) 170401 [2103.01232]

    work page arXiv 2022

  39. [39]

    de la Hamette, S.L

    A.-C. de la Hamette, S.L. Ludescher and M.P. Mueller, Entanglement-Asymmetry Correspondence for Internal Quantum Reference Frames,Phys. Rev. Lett.129 (2022) 260404 [2112.00046]

    work page arXiv 2022

  40. [40]

    Giacomini, Spacetime Quantum Reference Frames and superpositions of proper times, Quantum5, 508 (2021), arXiv:2101.11628 [quant-ph]

    F. Giacomini,Spacetime Quantum Reference Frames and superpositions of proper times,Quantum5(2021) 508 [2101.11628]

    work page arXiv 2021

  41. [41]

    Cepollaro, A

    C. Cepollaro, A. Akil, P. Cie´ sli´ nski, A.-C. de la Hamette and ˇC. Brukner,The sum of entanglement and subsystem coherence is invariant under quantum reference frame transformations,Phys. Rev. Lett.135 (2025) 010201 [2406.19448]

    work page arXiv 2025

  42. [42]

    The Perspectives of Non-Ideal Quantum Reference Frames

    S.C. Garmier, L. Hausmann and E. Castro-Ruiz,The perspectives of non-ideal quantum reference frames, [2512.19343]

    work page internal anchor Pith review Pith/arXiv arXiv

  43. [43]

    Song,Spinor analysis for quantum group suq(2), Journal of Physics A: Mathematical and General25 (1992) 2929

    X.-C. Song,Spinor analysis for quantum group suq(2), Journal of Physics A: Mathematical and General25 (1992) 2929

    1992

  44. [44]

    Castro-Ruiz, N

    E. Castro-Ruiz, N. Cohen, L. Barbado and ˇC. Brukner, Indefinite Probabilities from finite quantum reference frames,QRF 2025, Okinawa(16 th October 2025)

    2025

  45. [45]

    Castro-Ruiz, N

    E. Castro-Ruiz, N. Cohen, L. Barbado and ˇC. Brukner, Quantum Incompatibility of Relative Frequencies, Observers and Causality in Quantum Gravity, Bratislava(23 rd April 2026) . 9 Appendix A: Details of the computations In this appendix we present some technical computations performed to derive the results of the main text

    2026

  46. [46]

    (A1) where Rαβ γδ =q δ α γ δβ δ +ϵ αβϵγδ , ϵ αβ = 0 1 −q0 , ϵ αβ = 0−q −1 1 0 .(A2) andu α = x y ,w α = a c

    Details of the braiding relations As discussed in section III A, the braiding relations are defined by uα wβ =q −1Rαβ γδ wγ uδ , uα ¯wβ =R αβ γδ ¯wγ uδ . (A1) where Rαβ γδ =q δ α γ δβ δ +ϵ αβϵγδ , ϵ αβ = 0 1 −q0 , ϵ αβ = 0−q −1 1 0 .(A2) andu α = x y ,w α = a c . With these definitions, (A1) easily reads xa=ax , qxc−cx= q−q −1 ay , ay=qya , yc=cy xc∗ =qc ...

  47. [47]

    All orders commutator In this appendix, we present the expression, at all-orders in 1−q, for the commutator between two probability operatorsP i(↑),P j(↑) from which we explicitly derive the result (15). The commutator between two probabilities reads h Pi(↑), P j(↑) i = 1 q3 (1−q 2) −q 2ajc∗ j a∗ i ci +q 2ajc∗ j x∗ j yj −qa ∗ j ajc∗ i ci +q 3a∗ j ajc∗ i c...

  48. [48]

    Uncertainty principle for rotation matrix In this appendix, starting from (21), we prove that two observers, regardless of the rotation relating them, cannot sharply determine their relative orientation. The uncertainty principle between two generic entries of the rotation matrix reads ∆Rij∆Rkl ≥(1−q) (j+l)·(n i ×n k)−(n i +n k)·(j×l) .(A8) We recall that...