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arxiv: 2605.23862 · v1 · 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 2026-05-25 04:08 UTC · model grok-4.3

classification 🪐 quant-ph gr-qc
keywords indefinite probabilitiesquantum spacetimeSU_q(2) quantum groupspin measurementsnon-commuting operatorsrotational symmetryuncertainty principleStern-Gerlach apparatus
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The pith

Using the SU_q(2) quantum group for rotational symmetry makes spin-measurement probabilities 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 argues that in the low-energy limit of quantum gravity, the rotational symmetry of spin-1/2 systems and Stern-Gerlach apparatuses is described by the SU_q(2) quantum group. This description turns the probabilities of spin measurement outcomes into non-commuting operators rather than ordinary numbers. An uncertainty principle then arises between different probability operators, embodying a notion of indefinite probabilities. The same non-commutativity appears in the entries of the rotation matrix that relates two observers' reference frames, so the observers cannot determine their relative orientation with arbitrary sharpness.

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. This realizes a notion of indefinite probabilities reflected in an uncertainty principle between different probability operators. The non-commutativity is also present in the entries of the rotation matrix relating the reference frames of two observers, hence fundamentally preventing them from sharply measuring their relative orientation.

What carries the argument

The SU_q(2) quantum group, which deforms ordinary rotational symmetry so that the entries of the rotation matrix become non-commuting operators that carry the indefinite probabilities.

If this is right

  • An uncertainty relation holds between distinct probability operators for the outcomes of spin measurements.
  • The entries of any rotation matrix relating two observers' frames fail to commute.
  • Two observers cannot determine their relative orientation to arbitrary precision.
  • Probabilities themselves become indefinite quantities rather than definite numbers.

Where Pith is reading between the lines

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

  • The same non-commutativity could affect other measurement contexts that rely on rotational reference frames.
  • Precision spin experiments might eventually constrain the deformation parameter of the quantum group.
  • The construction supplies one concrete way in which quantum-spacetime effects could deepen the unpredictability already present in ordinary quantum mechanics.

Load-bearing premise

That the SU_q(2) quantum group supplies the correct description of rotational symmetry for spin-1/2 systems and Stern-Gerlach apparatuses in the low-energy limit of quantum gravity.

What would settle it

A direct calculation or measurement showing that the probability operators for spin outcomes commute with one another, or that the entries of the rotation matrix between two frames commute, would falsify the central claim.

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 / 2 minor

Summary. The manuscript claims that modeling rotational symmetry for spin-1/2 systems and Stern-Gerlach apparatuses via the SU_q(2) quantum group in the low-energy limit of quantum gravity yields non-commuting operators for the probabilities of spin-measurement outcomes. This produces an uncertainty relation among probability operators (indefinite probabilities) that is reflected in the non-commutativity of the entries of the rotation matrix relating two observers' frames, thereby preventing sharp determination of their relative orientation.

Significance. If the central modeling assumption is granted, the construction supplies a concrete algebraic realization of non-commuting probability operators and an associated uncertainty principle within a quantum-group framework. This could be of interest to the quantum-foundations and quantum-gravity communities as a formal extension of unpredictability. The result, however, inherits its physical content entirely from the choice of SU_q(2); absent an independent derivation of that symmetry from a concrete Planck-scale model, the significance remains conditional on an assumption that is standard in the quantum-group literature but not yet anchored in a specific effective theory of quantum spacetime.

major comments (2)
  1. [Abstract] Abstract and opening paragraphs: the manuscript takes the SU_q(2) quantum group as the appropriate rotational symmetry for spin-1/2 particles and Stern-Gerlach devices without deriving this choice from any concrete low-energy limit of a quantum-gravity model (e.g., a deformed Poincaré algebra or a specific regularization). Because the non-commutativity of the probability operators follows directly once SU_q(2) is adopted, the central claim that indefinite probabilities are realized is a modeling consequence rather than an emergent prediction; this assumption is therefore load-bearing for the entire argument.
  2. [Main text (derivation of uncertainty relation)] The uncertainty relation between probability operators is presented as a new physical feature, yet its derivation appears to rest on the built-in non-commutativity of the quantum-group rotation matrix. A explicit check that the same non-commutativity cannot be reproduced by an ordinary SU(2) rotation matrix (or by a different deformation) would strengthen the claim that the effect is distinctive to the quantum-group setting.
minor comments (2)
  1. Notation for the probability operators and their commutation relations should be introduced with an explicit equation number at first appearance to aid readability.
  2. The manuscript would benefit from a short paragraph contrasting the present construction with earlier uses of quantum groups in quantum gravity (e.g., references to deformed Poincaré algebras) to clarify the precise novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points about the scope of our modeling assumptions and the distinctiveness of the quantum-group effects. We address each major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and opening paragraphs: the manuscript takes the SU_q(2) quantum group as the appropriate rotational symmetry for spin-1/2 particles and Stern-Gerlach devices without deriving this choice from any concrete low-energy limit of a quantum-gravity model (e.g., a deformed Poincaré algebra or a specific regularization). Because the non-commutativity of the probability operators follows directly once SU_q(2) is adopted, the central claim that indefinite probabilities are realized is a modeling consequence rather than an emergent prediction; this assumption is therefore load-bearing for the entire argument.

    Authors: We agree that the choice of SU_q(2) is a foundational modeling assumption rather than a derivation from a specific Planck-scale regularization in this work. This choice is motivated by the established literature on quantum groups as effective symmetries in quantum spacetime (including deformed Poincaré algebras), and our aim is to derive the consequences for probability operators within that framework. We will revise the abstract and opening paragraphs to state this assumption more explicitly, cite the relevant quantum-group literature, and clarify that the indefinite probabilities are a consequence realized under this symmetry rather than an independent emergence from a full quantum-gravity model. revision: yes

  2. Referee: [Main text (derivation of uncertainty relation)] The uncertainty relation between probability operators is presented as a new physical feature, yet its derivation appears to rest on the built-in non-commutativity of the quantum-group rotation matrix. A explicit check that the same non-commutativity cannot be reproduced by an ordinary SU(2) rotation matrix (or by a different deformation) would strengthen the claim that the effect is distinctive to the quantum-group setting.

    Authors: We concur that an explicit comparison would clarify the role of the deformation. In the revised manuscript we will add a short calculation in the main text demonstrating that the standard SU(2) rotation matrix has commuting entries, leading to commuting probability operators and no uncertainty relation. This non-commutativity is therefore a direct consequence of the q-deformation. We will also briefly note that while other deformations could produce analogous effects, the SU_q(2) case is the standard one used for rotational symmetry in this context. revision: yes

Circularity Check

1 steps flagged

Non-commutativity of probabilities follows by construction from adopting SU_q(2) symmetry whose matrix elements are defined to be non-commuting

specific steps
  1. self definitional [Abstract]
    "we show that 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. 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"

    The non-commutativity of the rotation matrix entries is a built-in feature of the SU_q(2) quantum group definition. Therefore the claimed non-commuting probability operators, uncertainty principle, and 'indefinite probabilities' are equivalent to the modeling choice by construction; the paper presents this equivalence as a derived result.

full rationale

The paper's central result (indefinite probabilities realized via non-commuting probability operators and uncertainty principle) is obtained simply by choosing the SU_q(2) quantum group as the rotational symmetry; the non-commutativity is a defining property of that group rather than an independent consequence. The modeling choice itself is motivated by prior literature on quantum groups but not derived from a specific quantum-gravity model within the paper. This matches the self-definitional pattern where the claimed outcome reduces directly to the input ansatz.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the modeling assumption that SU_q(2) correctly captures rotational symmetry in this physical setting; no free parameters, invented entities, or additional axioms are identifiable from the abstract alone.

axioms (1)
  • domain assumption SU_q(2) quantum group describes rotational symmetry for spin-1/2 systems and Stern-Gerlach apparatuses
    Invoked directly in the abstract as the starting point for the construction.

pith-pipeline@v0.9.0 · 5650 in / 1382 out tokens · 44439 ms · 2026-05-25T04:08:37.952687+00:00 · methodology

discussion (0)

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    (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

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    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...