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arxiv: 2606.08704 · v1 · pith:4NDK4PEVnew · submitted 2026-06-07 · 🪐 quant-ph

Regularised Arbitrary Gauge non-Relativistic QED

Alex Chivers-White , Adam Stokes This is my paper

Pith reviewed 2026-06-27 18:26 UTC · model grok-4.3

classification 🪐 quant-ph
keywords non-relativistic QEDarbitrary gaugeregularizationmultipolar gaugeCoulomb gaugeperturbation theoryDicke modelelectric dipole approximation
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The pith

A regularised arbitrary-gauge formulation of nonrelativistic QED allows direct comparison of Coulomb and multipolar descriptions.

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

The authors construct a version of nonrelativistic quantum electrodynamics that incorporates regularization while remaining valid in any gauge choice. They apply this construction to contrast the Coulomb gauge with the multipolar gauge when both use a Lorentzian form factor to cut off high frequencies. The analysis covers how regularization changes results in perturbation theory for different Hamiltonian partitions and tests the electric dipole approximation. In the multipolar gauge the regularization produces a trade-off that grows with the cut-off and reduces direct interactions between separate atoms.

Core claim

We develop a regularised arbitrary-gauge formulation of nonrelativistic quantum electrodynamics and use it to compare Coulomb and multipolar descriptions with a Lorentzian form factor. We analyse the effect of regularisation in perturbation theory, including alternative partitions of the Hamiltonian into free and interaction parts, and the limits of the electric dipole approximation. The regularised multipolar gauge exhibits a cut-off-dependent trade-off between the strength of individual interaction terms and the localisation of material subsystems that suppresses direct inter-atomic interactions.

What carries the argument

The regularised arbitrary-gauge formulation of nonrelativistic QED, built around a Lorentzian form factor that preserves gauge invariance while enabling controlled comparisons between different gauge choices.

If this is right

  • The multipolar gauge shows a cut-off-dependent trade-off that weakens direct inter-atomic interactions through increased localisation of material subsystems.
  • Regularisation affects the strength of interaction terms in perturbation theory depending on how the Hamiltonian is partitioned.
  • The framework applies to short-range phenomena including Dicke criticality.
  • The electric dipole approximation has defined limits once regularization is included.

Where Pith is reading between the lines

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

  • This formulation could be used to check gauge independence in numerical simulations of light-matter systems.
  • The observed trade-off may alter predictions for collective effects in dense atomic ensembles.
  • Extensions might explore other form factors to see if the trade-off persists.

Load-bearing premise

The Lorentzian form factor remains suitable when the formulation is extended to arbitrary gauges for the purpose of comparing descriptions and extracting the cut-off trade-off.

What would settle it

An explicit calculation in the regularised multipolar gauge that fails to exhibit the cut-off-dependent trade-off between interaction strength and subsystem localisation would contradict the reported analysis.

Figures

Figures reproduced from arXiv: 2606.08704 by Adam Stokes, Alex Chivers-White.

Figure 1
Figure 1. Figure 1: FIG. 1: Energy/length scale diagram. The horizontal axis [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Main [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The ratios [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The ratio of the regularised transition element to the [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

We develop a regularised arbitrary-gauge formulation of nonrelativistic quantum electrodynamics and use it to compare Coulomb and multipolar descriptions with a Lorentzian form factor. We analyse the effect of regularisation in perturbation theory, including alternative partitions of the Hamiltonian into free and interaction parts, and the limits of the electric dipole approximation. The regularised multipolar gauge exhibits a cut-off-dependent trade-off between the strength of individual interaction terms and the localisation of material subsystems that suppresses direct inter-atomic interactions. We discuss the implications of the framework for short-range phenomena, including Dicke criticality.

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

0 major / 2 minor

Summary. The manuscript develops a regularised arbitrary-gauge formulation of nonrelativistic quantum electrodynamics (NRQED) employing a Lorentzian form factor. It compares the Coulomb and multipolar gauges, analyses the impact of regularisation within perturbation theory (including alternative partitions of the Hamiltonian into free and interaction parts), and examines the limits of the electric dipole approximation. A key result is that the regularised multipolar gauge exhibits a cut-off-dependent trade-off between interaction-term strength and material-subsystem localisation that suppresses direct inter-atomic interactions; implications for short-range phenomena such as Dicke criticality are discussed.

Significance. If the central derivations hold, the framework supplies a systematic way to compare gauges under consistent regularisation, which is useful for short-range light-matter problems where gauge artefacts and cut-off dependence can be pronounced. The explicit treatment of alternative Hamiltonian partitions and the localisation trade-off constitute concrete, falsifiable predictions that could be tested in few-atom or cavity-QED settings.

minor comments (2)
  1. The abstract states that a Lorentzian form factor is used but does not specify its explicit functional form or the precise manner in which it is inserted into the arbitrary-gauge vector potential; adding this definition early would aid readability.
  2. No numerical values or plots illustrating the cut-off-dependent trade-off are referenced in the provided summary; if such results exist in the manuscript, a brief mention of the relevant figure or table would strengthen the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of our work on a regularised arbitrary-gauge formulation of NRQED. The report provides an accurate overview of the manuscript's scope and notes its potential utility, but lists no specific major comments under the MAJOR COMMENTS section. We therefore provide no point-by-point responses.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract and description outline a constructive development of a regularised arbitrary-gauge NRQED formulation using a Lorentzian form factor, followed by perturbative analysis and comparisons of gauges. No derivation chain, equations, or self-citations are exhibited that reduce a claimed prediction or result to its inputs by construction. The regularization scheme is presented as an adopted tool for comparison rather than a self-referential fit or renamed known result. Without load-bearing steps that match the enumerated circularity patterns, the framework remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be identified beyond the implicit choice of Lorentzian form factor as a regularization tool.

free parameters (1)
  • cut-off
    Referenced via cut-off-dependent trade-off in multipolar gauge; appears to be a regularization scale parameter.

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Reference graph

Works this paper leans on

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    (51) that results from the choices φ .= 1 and gT = 0

    Coulomb gauge The natural definition for comparison is the well- known material Hamiltonian in Eq. (51) that results from the choices φ .= 1 and gT = 0. In units such that q2/(4πϵ0) = 1 the standard Coulomb potential therein is UC(r) = −1/r (ignoring the infinite self-term). More generally, a prototypical (spherically symmetric) choice of form factor is t...

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    Multipolar gauges The standard multipolar gauge choice ofgT in Eq. (10) yields, in the point charge limit kφ → ∞ (with q2/(4πϵ0) = 1), δUM(r) = 2π Z 1 0 dλdλ′r · δT([λ − λ′]r) · r. (56) 6 The integrand is singular at contact points λ = λ′. A finite term, δUMℓ, is obtained in the point charge limit for the regularised multipolar-gauge choice gℓT(x, x′) = −...

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    Coulomb gauge In the Coulomb gauge (gT = 0), Eq. (47) is simply the overlap of the electrostatic fields of the two atoms UCF12 = Z d3x EF L1(x) · EF L2(x) = Z d3xd3x′ ρF 1(x)ρF 2(x′) 4π|x − x′| . (92) For a relativistic cut-off kM ∼ λ−1 c this interaction is es- sentially indistinguishable from the bar electrostatic in- teraction. In the EDA one obtains t...

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    To be precise, changing units by reinserting a factor of 16 q2/(4πϵ0) yields the result given in Ref. [30], namely, δUMℓ(r) = k3 ℓ d2/(24πϵ0)