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arxiv: 2504.20218 · v2 · submitted 2025-04-28 · 🪐 quant-ph · math-ph· math.MP

Fractional Angular Momentum and Quasi-Probability Densities for Angular Degrees of Freedom

Bo-Sture K. Skagerstam , Per K. Rekdal This is my paper

Pith reviewed 2026-05-22 17:18 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords fractional angular momentumquasi-probability densitiesangular positionangular momentum operatorBorn's rulequantum witnessesuncertainties
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The pith

Properly defined two-parameter quasi-probability densities serve as witnesses for quantum behaviour in pure states involving angular position and angular momentum L, where negatives flag fractional mean L superpositions ambiguously.

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

The paper examines two-parameter quasi-probability densities for systems described by a self-adjoint angular position observable and the angular momentum operator L. It demonstrates that these densities can exhibit negative values for superpositions of angular momentum eigenstates possessing a fractional mean value of L, potentially revealing their quantum nature though in an ambiguous manner. For certain choices of parameters, the densities remain positive and align with the probabilities given by Born's rule in quantum mechanics. Additionally, measurements of the uncertainties in the angular position and L observables can be enough to uncover distinctive quantum-mechanical aspects of these states, without requiring the quasi-probability densities.

Core claim

In the present update work we consider properly defined two-parameter quasi-probability densities that, e.g., can be used as witness for quantum behaviour for a class of pure states as expressed in terms of a self-adjoint observable angular position and the corresponding angular momentum operator L. It is shown that negative values of the corresponding quasi-probability densities may reveal the quantum nature of superpositions of angular momentum eigenstates with a fractional mean value of L but in an ambiguous manner. For a suitable choice of parameters these quasi-probability densities are positive and are in accordance with Borns rule in quantum mechanics. It is also shown that experiment

What carries the argument

The two-parameter quasi-probability densities for angular position and angular momentum observables that can take negative values for certain superpositions or stay positive for suitable parameters.

If this is right

  • Negative values of the quasi-probability densities may reveal the quantum nature of superpositions with fractional mean angular momentum, but only in an ambiguous manner.
  • A suitable choice of parameters makes the quasi-probability densities positive and consistent with Born's rule.
  • Experimental data on the uncertainties for angular position and L observables can reveal unique quantum-mechanical features without using quasi-probability densities.

Where Pith is reading between the lines

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

  • The ambiguity in using negative values as a quantum witness suggests that uncertainties provide a more direct experimental route for angular systems.
  • These densities could be compared against other quasi-probability constructions like the Wigner function to test consistency across different parameter regimes.
  • If uncertainties alone suffice, then similar uncertainty-based tests might apply to other pairs of conjugate observables with periodic boundary conditions.

Load-bearing premise

The two-parameter quasi-probability densities are properly defined such that they can serve as witnesses for quantum behaviour in the class of pure states considered.

What would settle it

Preparing a superposition state with fractional mean angular momentum, computing or measuring the two-parameter quasi-probability density, and checking whether it shows negative values or stays positive under chosen parameters while matching Born rule probabilities.

Figures

Figures reproduced from arXiv: 2504.20218 by Bo-Sture K. Skagerstam, Per K. Rekdal.

Figure 1
Figure 1. Figure 1: The Wigner distribution W[θ, p] according to Eq.(15) with −π ≤ θ ≤ π and ¯θ = 0 for the state ψ(θ) as given in Eq.(3). In general W[θ, p] is bounded by |W[θ, p]| ≤ 1/2π . Here q = exp(−1/2λ) = 0.5 and integer-valued ¯l = l as a function of the parameter m = p−l in the range −2 ≤ m ≤ 2. With this choice of parameters W[θ, p] is then symmetric with respect to a line with m = 0. For half-integer values of m t… view at source ↗
Figure 2
Figure 2. Figure 2: The Wigner distribution W[θ, p] according to Eq.(15) for −π ≤ θ ≤ π as in Fig.1 now with q = exp(−1/2λ) = 0.001, ϵ = 1/2, and ¯l = l + 1/2 in the range −2 ≤ m ≤ 2 with m = p−l for the state ψ(θ) as in Eq.(3). The symmetry with respect to the line with m = 0 as in Fig.1 is now not present. For half-integer values of m the Wigner function W[θ, p] can be negative at |θ| = O(π) and the parameter p must then be… view at source ↗
Figure 3
Figure 3. Figure 3: The Wigner distribution W1/2[θ, p] according to Eq.(34) for −π ≤ θ ≤ π for −π ≤ θ ≤ π with q = 0.001, ¯l = l + ϵ, and ϵ = 1/2 for the state ψ(θ) according to Eq.(3) as a function of the parameter m = p − l in the range −2 ≤ m ≤ 2. As in Fig.2 the symmetry with respect to the line with m = 0 is absent. Regions of negativity for W1/2[θ, p] are in general not the same as in Fig.2 for the same quantum state co… view at source ↗
Figure 4
Figure 4. Figure 4: The upper solid curve corresponds to the uncertainty ∆ [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
read the original abstract

In the present update work we consider properly defined two-parameter quasi-probability densities that, e.g., can be used as witness for quantum behaviour for a class of pure states as expressed in terms of a self-adjoint observable angular position and the corresponding angular momentum operator L. It is shown that negative values of the corresponding quasi-probability densities may reveal the quantum nature of superpositions of angular momentum eigenstates with a fractional mean value of L but in an ambiguous manner. For a suitable choice of parameters these quasi-probability densities are positive and are in accordance with Borns rule in quantum mechanics. It is also shown that experimental data of the uncertainties for the angular position and L observables can be sufficient to reveal some unique quantum-mechanical features of such states without necessarily making use of quasi-probability densities.

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 / 1 minor

Summary. The manuscript presents an update on two-parameter quasi-probability densities for angular degrees of freedom, specifically for a self-adjoint angular position observable and the angular momentum operator L. It claims that negative values of these densities can indicate the quantum nature of superpositions of angular momentum eigenstates with fractional mean L, albeit ambiguously. For appropriate parameter choices, the densities are positive and consistent with Born's rule. Additionally, it suggests that experimental uncertainties in angular position and L can suffice to reveal unique quantum features without relying on quasi-probability densities.

Significance. If the claims hold, this work could advance the understanding of quasi-probability representations in systems with angular degrees of freedom, particularly highlighting the role of fractional angular momentum and providing uncertainty-based methods to detect quantum features. However, the abstract-only availability limits a full assessment of the novelty and impact.

major comments (2)
  1. [Abstract] The central claims depend on the proper definition of the two-parameter quasi-probability densities, but without access to the full manuscript, including mathematical definitions, derivations, and any supporting evidence or examples, it is not possible to evaluate whether these densities are rigorously defined or if the negativity indeed serves as an ambiguous witness for quantum behavior in the described states.
  2. [Abstract] The statement that 'experimental data of the uncertainties for the angular position and L observables can be sufficient to reveal some unique quantum-mechanical features' lacks any indication of the specific features or how the data suffices, making it difficult to assess the strength of this claim without further details.
minor comments (1)
  1. [Abstract] The abstract mentions 'Borns rule' which should be 'Born's rule' for grammatical correctness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their comments on our manuscript. We address the major comments point by point below. The full manuscript on arXiv:2504.20218 contains the detailed mathematical definitions, derivations, and examples referenced in the abstract; we regret that only the abstract appears to have been available for review and are happy to expand the abstract or add clarifications as needed.

read point-by-point responses

  1. Referee: [Abstract] The central claims depend on the proper definition of the two-parameter quasi-probability densities, but without access to the full manuscript, including mathematical definitions, derivations, and any supporting evidence or examples, it is not possible to evaluate whether these densities are rigorously defined or if the negativity indeed serves as an ambiguous witness for quantum behavior in the described states.

    Authors: The full manuscript defines the two-parameter quasi-probability densities rigorously for the self-adjoint angular position observable and the angular momentum operator L, with explicit constructions, derivations of their properties, and concrete examples for superpositions of angular momentum eigenstates having fractional mean L. These examples illustrate how negativity can serve as an (ambiguous) witness for quantum behavior. We can revise the abstract to include a concise statement of the key definition if that addresses the concern. revision: partial

  2. Referee: [Abstract] The statement that 'experimental data of the uncertainties for the angular position and L observables can be sufficient to reveal some unique quantum-mechanical features' lacks any indication of the specific features or how the data suffices, making it difficult to assess the strength of this claim without further details.

    Authors: The full manuscript shows that measured uncertainties in angular position and L for the indicated states can violate classical bounds or exhibit signatures tied to the fractional mean angular momentum that are impossible for classical mixtures, thereby revealing quantum features directly from uncertainty data. We will revise the abstract to specify these features and the manner in which the data suffices. revision: yes

Circularity Check

0 steps flagged

No significant circularity detectable from abstract

full rationale

Only the abstract is provided, which states results on two-parameter quasi-probability densities for angular position and L without any equations, derivation steps, self-citations, or fitted parameters. No load-bearing claim reduces by construction to its inputs, as no specific mathematical chain or ansatz is exhibited. The assertions that negativity may reveal quantum features (ambiguously) and that suitable parameters yield positivity consistent with Born's rule are presented as demonstrated outcomes, but the absence of the underlying definitions or proofs means no circularity of any enumerated kind can be identified or quoted. The paper's claims appear self-contained within the given text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no details on free parameters, axioms, or invented entities; full text required to identify any fitted values, background assumptions, or new postulated constructs.

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Reference graph

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