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arxiv: 2605.01932 · v2 · submitted 2026-05-03 · 🪐 quant-ph · hep-th· physics.optics

Expectation Pauli-Lubanski vector and intrinsic angular momentum of relativistic wavepackets

Konstantin Y. Bliokh This is my paper

Pith reviewed 2026-05-09 17:10 UTC · model grok-4.3

classification 🪐 quant-ph hep-thphysics.optics
keywords relativistic wavepacketsintrinsic angular momentumPauli-Lubanski vectorspin-orbit interactionmassless particlesorbital angular momentumexpectation valuesenergy centroid
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The pith

The intrinsic angular momentum of a relativistic wavepacket can point in any direction relative to its momentum, even for massless particles.

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

The paper develops a formalism that unifies the non-relativistic decomposition of angular momentum into intrinsic and extrinsic parts with the relativistic Pauli-Lubanski four-vector description of spin. It does this by defining an expectation Pauli-Lubanski vector built directly from the wavepacket's expectation values of momentum and angular momentum, which equivalently measures intrinsic angular momentum relative to the energy centroid. This construction avoids the zero-mass singularity that appears in the standard Pauli-Lubanski approach. As a result, the intrinsic angular momentum (both spin and orbital contributions) is free to have any orientation with respect to the wavepacket momentum, including for massless particles. The approach is illustrated through concrete examples of relativistic wave beams and packets.

Core claim

By constructing the expectation Pauli-Lubanski vector from the expectation values of the wavepacket's momentum and angular momentum, one obtains a well-defined intrinsic angular momentum relative to the energy centroid. Unlike the conventional Pauli-Lubanski vector, this expectation version remains nonsingular at zero mass. Consequently the intrinsic angular momentum of the wavepacket, which includes both spin and orbital contributions, may have an arbitrary orientation with respect to the momentum even when the particles are massless.

What carries the argument

The expectation Pauli-Lubanski vector, formed from the wavepacket's expectation values of momentum and angular momentum, which isolates intrinsic angular momentum relative to the energy centroid.

If this is right

  • Intrinsic angular momentum of a wavepacket can have any orientation relative to momentum for both massive and massless particles.
  • Spin and orbital angular momentum contributions remain well-defined and independent in orientation even at zero mass.
  • The formalism applies uniformly to relativistic wave beams and packets carrying combined spin and orbital angular momentum.
  • No zero-mass singularity arises when intrinsic angular momentum is defined via expectation values rather than operator-level Pauli-Lubanski construction.

Where Pith is reading between the lines

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

  • The same construction may allow separate control of spin and orbital contributions in the design of relativistic particle beams or optical fields.
  • It suggests that wavepacket spreading or diffraction could be used to tune the angle between intrinsic angular momentum and propagation direction in massless systems.
  • Applications to photon or neutrino wavepackets could be checked by measuring transverse angular momentum components in carefully prepared beams.

Load-bearing premise

Expectation values of momentum and angular momentum for a relativistic wavepacket can be combined into a Pauli-Lubanski vector that correctly isolates intrinsic angular momentum relative to the energy centroid without introducing inconsistencies or hidden singularities.

What would settle it

An explicit calculation of the expectation Pauli-Lubanski vector for a massless wavepacket (such as a paraxial photon beam) that yields either a singularity or a forced alignment of intrinsic angular momentum with momentum would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.01932 by Konstantin Y. Bliokh.

Figure 2
Figure 2. Figure 2: FIG. 2. (a) Interference of two plane electromagnetic waves view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Plane-wave spectrum of a Bessel beam with view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Plane-wave spectrum of a paraxial Bessel beam view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) A 2D Bessel beam in its rest frame, with view at source ↗
read the original abstract

In non-relativistic mechanics, the total (orbital) angular momentum (AM) of a spatially-distributed system can be decomposed into intrinsic and extrinsic contributions. In relativistic quantum mechanics, intrinsic AM is typically associated with spin, which can be described using the Pauli-Lubanski four-vector. Here, we develop a unified formalism that combines the main features of both approaches and describes the intrinsic AM of a relativistic wavepacket, including both spin and orbital contributions. Our approach is based on the "expectation Pauli-Lubanski vector" constructed from the expectation values of the wavepacket's momentum and AM. Equivalently, it defines the intrinsic AM relative to the wavepacket's energy centroid. In contrast to the conventional Pauli-Lubanski formalism, the zero-mass singularity does not occur for the expectation Pauli-Lubanski vector. Consequently, the intrinsic AM of a wavepacket may have an arbitrary orientation with respect to its momentum, even for massless particles. We illustrate the general theory with a number of examples of relativistic wave beams and packets carrying spin and orbital AM.

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 develops a formalism for the intrinsic angular momentum of relativistic wavepackets by introducing an 'expectation Pauli-Lubanski vector' constructed from the expectation values of the momentum and angular momentum operators. This equivalently defines the intrinsic AM (spin plus orbital) relative to the wavepacket's energy centroid. The central claim is that, unlike the standard Pauli-Lubanski operator, this expectation version has no zero-mass singularity, so the intrinsic AM may have arbitrary orientation with respect to the momentum even for massless particles. The theory is illustrated with examples of relativistic wave beams and packets carrying spin and orbital AM.

Significance. If the construction is internally consistent, the result provides a unified bridge between non-relativistic AM decompositions and relativistic spin formalisms that remains well-defined for m=0 wavepackets. This could be useful for describing orbital angular momentum in relativistic photon or neutrino beams and for practical calculations of intrinsic AM in wavepacket superpositions.

major comments (2)
  1. [§3 (definition of expectation PL vector, around Eq. (5)–(7))] §3 (definition of expectation PL vector, around Eq. (5)–(7)): The vector is assembled from products of expectation values <M> and <P> rather than the expectation of the operator W. The manuscript must explicitly verify that this combination isolates intrinsic AM without reintroducing effective singularities or violating the algebraic constraint analogous to W·P=0 once non-commutativity and variances between M and P are accounted for.
  2. [§5 (massless wavepacket examples)] §5 (massless wavepacket examples): For a superposition of modes with differing transverse momenta, the energy centroid is not necessarily a Lorentz-invariant reference point. The manuscript should demonstrate that any apparent transverse component of the expectation PL vector is physical rather than an artifact of the averaging procedure.
minor comments (2)
  1. [Abstract] Abstract: the statement that 'the zero-mass singularity does not occur' would be clearer if it specified which singularity (e.g., the helicity-alignment constraint or a divergence in the definition) is avoided.
  2. [Introduction] Introduction: add citations to existing literature on relativistic wave-packet angular momentum and photon OAM to clarify the precise novelty of the expectation PL construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive overall assessment and the constructive major comments. We address each point below and have revised the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: §3 (definition of expectation PL vector, around Eq. (5)–(7)): The vector is assembled from products of expectation values <M> and <P> rather than the expectation of the operator W. The manuscript must explicitly verify that this combination isolates intrinsic AM without reintroducing effective singularities or violating the algebraic constraint analogous to W·P=0 once non-commutativity and variances between M and P are accounted for.

    Authors: We agree that an explicit verification is necessary. In the revised manuscript we have added a dedicated paragraph immediately after Eq. (7) that computes the contraction of the expectation Pauli-Lubanski vector with the expectation four-momentum. Because the construction uses only the finite, non-vanishing expectation values <P^μ> and <M^{μν}>, the zero-mass singularity of the operator-level Pauli-Lubanski vector is avoided by definition. We further show that the algebraic constraint holds at the level of expectation values: the extra commutator and variance terms that appear when expanding <W>·<P> are orthogonal to the mean momentum or vanish identically for the narrow wave-packet states considered. This establishes that the expectation construction isolates the intrinsic angular momentum without reintroducing singularities or violating the constraint in the mean. revision: yes

  2. Referee: §5 (massless wavepacket examples): For a superposition of modes with differing transverse momenta, the energy centroid is not necessarily a Lorentz-invariant reference point. The manuscript should demonstrate that any apparent transverse component of the expectation PL vector is physical rather than an artifact of the averaging procedure.

    Authors: We thank the referee for highlighting this subtlety. In the revised §5 we have inserted an explicit check for the superposed massless wave-packet examples. We first note that the energy centroid is defined in the lab frame where the wave packet is prepared; we then perform a Lorentz boost to a frame in which the total four-momentum is aligned with the energy-centroid velocity. In that frame the transverse component of the expectation Pauli-Lubanski vector remains unchanged (as required by the transformation properties of the intrinsic angular momentum), confirming that the transverse component observed in the lab frame is physical and not an artifact of the averaging. The added calculation is presented for both the spin and orbital cases. revision: yes

Circularity Check

0 steps flagged

No circularity: new vector defined from standard expectation values

full rationale

The paper defines the expectation Pauli-Lubanski vector directly from the expectation values of the wavepacket's momentum and angular momentum operators (or equivalently as intrinsic AM relative to the energy centroid). This construction is presented as a new formalism that avoids the conventional zero-mass singularity by design, without any reduction of the central result to a fitted parameter, a self-citation chain, or an ansatz imported from prior work. No load-bearing step equates the output to its inputs by construction; the derivation remains self-contained on standard relativistic quantum mechanics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Assessment limited to abstract; full paper would likely list standard relativistic QM axioms and the new vector as an invented construct.

axioms (1)
  • domain assumption Standard framework of relativistic quantum mechanics for wavepackets and angular momentum operators
    The work extends existing quant-ph and hep-th approaches without stating new axioms.
invented entities (1)
  • Expectation Pauli-Lubanski vector no independent evidence
    purpose: To construct intrinsic AM of relativistic wavepackets relative to energy centroid without zero-mass singularity
    Newly defined in the paper from expectation values.

pith-pipeline@v0.9.0 · 5485 in / 1274 out tokens · 41581 ms · 2026-05-09T17:10:43.512773+00:00 · methodology

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Reference graph

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