Regularised Multipolar Gauge

• Regularised multipolar gauge is a framework ensuring that truncated minimal-coupling Hamiltonians and multipolar Hamiltonians remain gauge-equivalent beyond the electric dipole approximation.
• It methodically incorporates finite-order truncation, ultraviolet smearing, and many-body regularisation to control divergences and ill-defined operators.
• The approach also extends to topological phases via higher-form gauge theories, balancing charge localization and intersystem interactions under compact conditions.

Regularised multipolar gauge denotes a family of gauge-theoretic constructions in which a multipolar, or length-type, description of light–matter coupling is made well defined under truncation, ultraviolet regularisation, periodic boundary conditions, or lattice compactification. In the literature considered here, the term appears in several technically distinct but related senses: as the finite-order multipolar gauge uniquely gauge-equivalent to a truncated minimal-coupling Hamiltonian beyond the electric dipole approximation; as the Poincaré, or multipolar, gauge supplemented by ultraviolet smearing in nonrelativistic QED; as a many-body regularisation of electric multipole operators under periodic boundary conditions; and as a lattice-regularised higher-form gauge description of multipole topological phases (Anzaki et al., 2018, Stokes et al., 2021, Wheeler et al., 2018, Chivers-White et al., 7 Jun 2026, Dubinkin et al., 2020). Across these settings, the unifying theme is that the multipolar representation is not introduced ad hoc, but derived in a way that preserves the relevant notion of gauge consistency.

1. Exact multipolar gauge and its identification with Poincaré gauge

In nonrelativistic strong-field theory, the starting point is the minimal-coupling velocity-gauge Hamiltonian in Coulomb gauge,

$$H_{\rm VG} = \frac{[- i \nabla - q\mathbf{A}(\mathbf{r},t)]^2}{2M},$$

with $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$ and $$\mathbf{B}(\mathbf{x},t) = \nabla_x \times \mathbf{A}(\mathbf{x},t)$$. A Power–Zienau–Woolley-type transformation with

$$W = e^{i q\chi(\mathbf{r},t)},\qquad \chi(\mathbf{r},t) = \int_0^1 \mathbf{r}\cdot \mathbf{A}(\lambda\mathbf{r},t)\, d\lambda$$

yields the exact multipolar Hamiltonian

$$H_{\rm LG} = \frac{1}{2M}\left[ - i \nabla + q\int_0^1 \lambda\mathbf{r}\times \mathbf{B}(\lambda\mathbf{r},t)\, d\lambda \right]^2 - q\int_0^1 \mathbf{r}\cdot\mathbf{E}(\lambda\mathbf{r},t)\, d\lambda,$$

with $\psi_{\rm LG} = W^{-1}\psi_{\rm VG}$ (Anzaki et al., 2018). This is the exact, all-orders multipolar gauge.

Within constrained nonrelativistic QED, the same multipolar structure appears as the Poincaré gauge, defined by

$\mathbf x\cdot\mathbf A(\mathbf x)=0.$

Stokes and Nazir show, using Dirac’s constrained quantisation, that Poincaré-gauge QED and multipolar, or Power–Zienau–Woolley, nonrelativistic QED are identical, and both are unitarily equivalent to Coulomb-gauge QED. The apparent incompatibility in earlier debates is traced to a semantic mismatch concerning “canonical momentum”: in the multipolar formulation the canonical pair is $(\mathbf A_{\mathrm T},\bm\Pi)$ with $\bm\Pi=-\mathbf D_{\mathrm T}$, not $(\mathbf A_{\mathrm T},-\mathbf E_{\mathrm T})$ (Stokes et al., 2021).

This identification fixes the formal status of the multipolar gauge. In the Poincaré choice,

$$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$0

which is the standard multipolar polarization density of an atom, and the Hamiltonian can be written in the multipolar canonical variables as

$$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$1

with $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$2 in Poincaré gauge (Stokes et al., 2021). A plausible implication is that any regularised multipolar gauge in nrQED must preserve this unitary gauge-fixing structure rather than alter the gauge concept itself.

2. Finite-order regularised multipolar gauge beyond the electric dipole approximation

The most explicit finite-order construction is given in the analysis of gauge invariance beyond the electric dipole approximation. The vector potential is Taylor expanded about $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$3 and truncated at order $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$4,

$$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$5

leading to

$$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$6

All dependence on $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$7 is kept exactly, including the nonlinear term

$$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$8

which is emphasized as conceptually essential for gauge invariance at fixed order (Anzaki et al., 2018).

The corresponding truncated multipolar Hamiltonian is defined by truncating $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$9 to order $$\mathbf{B}(\mathbf{x},t) = \nabla_x \times \mathbf{A}(\mathbf{x},t)$$0 and $$\mathbf{B}(\mathbf{x},t) = \nabla_x \times \mathbf{A}(\mathbf{x},t)$$1 to order $$\mathbf{B}(\mathbf{x},t) = \nabla_x \times \mathbf{A}(\mathbf{x},t)$$2,

$$\mathbf{B}(\mathbf{x},t) = \nabla_x \times \mathbf{A}(\mathbf{x},t)$$3

and the truncated gauge transformation is generated by

$$\mathbf{B}(\mathbf{x},t) = \nabla_x \times \mathbf{A}(\mathbf{x},t)$$4

The central result is

$$\mathbf{B}(\mathbf{x},t) = \nabla_x \times \mathbf{A}(\mathbf{x},t)$$5

so the unique gauge partner of $$\mathbf{B}(\mathbf{x},t) = \nabla_x \times \mathbf{A}(\mathbf{x},t)$$6 is $$\mathbf{B}(\mathbf{x},t) = \nabla_x \times \mathbf{A}(\mathbf{x},t)$$7 (Anzaki et al., 2018).

The multipole-expanded form of $$\mathbf{B}(\mathbf{x},t) = \nabla_x \times \mathbf{A}(\mathbf{x},t)$$8 contains electric multipoles up to E$$\mathbf{B}(\mathbf{x},t) = \nabla_x \times \mathbf{A}(\mathbf{x},t)$$9, magnetic multipoles up to M$$W = e^{i q\chi(\mathbf{r},t)},\qquad \chi(\mathbf{r},t) = \int_0^1 \mathbf{r}\cdot \mathbf{A}(\lambda\mathbf{r},t)\, d\lambda$$0, and two additional terms: $$W = e^{i q\chi(\mathbf{r},t)},\qquad \chi(\mathbf{r},t) = \int_0^1 \mathbf{r}\cdot \mathbf{A}(\lambda\mathbf{r},t)\, d\lambda$$1 and

$$W = e^{i q\chi(\mathbf{r},t)},\qquad \chi(\mathbf{r},t) = \int_0^1 \mathbf{r}\cdot \mathbf{A}(\lambda\mathbf{r},t)\, d\lambda$$2

These are the time-derivative multipole term and the nonlinear magnetic term. Both are usually neglected in heuristic treatments, but they are shown to be indispensable for strict gauge equivalence with $$W = e^{i q\chi(\mathbf{r},t)},\qquad \chi(\mathbf{r},t) = \int_0^1 \mathbf{r}\cdot \mathbf{A}(\lambda\mathbf{r},t)\, d\lambda$$3 (Anzaki et al., 2018).

For $$W = e^{i q\chi(\mathbf{r},t)},\qquad \chi(\mathbf{r},t) = \int_0^1 \mathbf{r}\cdot \mathbf{A}(\lambda\mathbf{r},t)\, d\lambda$$4, the first beyond-dipole regularised multipolar gauge is $$W = e^{i q\chi(\mathbf{r},t)},\qquad \chi(\mathbf{r},t) = \int_0^1 \mathbf{r}\cdot \mathbf{A}(\lambda\mathbf{r},t)\, d\lambda$$5, containing E1, M1, E2, and the nonlinear magnetic term: $$W = e^{i q\chi(\mathbf{r},t)},\qquad \chi(\mathbf{r},t) = \int_0^1 \mathbf{r}\cdot \mathbf{A}(\lambda\mathbf{r},t)\, d\lambda$$6 This Hamiltonian, and not a truncated E1+M1 form missing the quadratic magnetic term, is gauge-equivalent to $$W = e^{i q\chi(\mathbf{r},t)},\qquad \chi(\mathbf{r},t) = \int_0^1 \mathbf{r}\cdot \mathbf{A}(\lambda\mathbf{r},t)\, d\lambda$$7 (Anzaki et al., 2018).

The same work proves that gauge equivalence holds only when truncation orders are matched, $$W = e^{i q\chi(\mathbf{r},t)},\qquad \chi(\mathbf{r},t) = \int_0^1 \mathbf{r}\cdot \mathbf{A}(\lambda\mathbf{r},t)\, d\lambda$$8, and that the alternative truncation $$W = e^{i q\chi(\mathbf{r},t)},\qquad \chi(\mathbf{r},t) = \int_0^1 \mathbf{r}\cdot \mathbf{A}(\lambda\mathbf{r},t)\, d\lambda$$9, obtained by expanding $$H_{\rm LG} = \frac{1}{2M}\left[ - i \nabla + q\int_0^1 \lambda\mathbf{r}\times \mathbf{B}(\lambda\mathbf{r},t)\, d\lambda \right]^2 - q\int_0^1 \mathbf{r}\cdot\mathbf{E}(\lambda\mathbf{r},t)\, d\lambda,$$0 itself to order $$H_{\rm LG} = \frac{1}{2M}\left[ - i \nabla + q\int_0^1 \lambda\mathbf{r}\times \mathbf{B}(\lambda\mathbf{r},t)\, d\lambda \right]^2 - q\int_0^1 \mathbf{r}\cdot\mathbf{E}(\lambda\mathbf{r},t)\, d\lambda,$$1, has no clean multipolar gauge partner and yields an incorrect Lorentz force (Anzaki et al., 2018). In this finite-order sense, “regularised multipolar gauge” means an order-by-order multipolar Hamiltonian derived, rather than guessed, from a truncated minimal-coupling Hamiltonian.

3. Ultraviolet regularisation in arbitrary-gauge nonrelativistic QED

A distinct use of the term arises in regularised arbitrary-gauge nrQED. In that setting, point-charge matter densities are smeared with a spherically symmetric form factor $$H_{\rm LG} = \frac{1}{2M}\left[ - i \nabla + q\int_0^1 \lambda\mathbf{r}\times \mathbf{B}(\lambda\mathbf{r},t)\, d\lambda \right]^2 - q\int_0^1 \mathbf{r}\cdot\mathbf{E}(\lambda\mathbf{r},t)\, d\lambda,$$2, chosen in momentum space as a Lorentzian,

$$H_{\rm LG} = \frac{1}{2M}\left[ - i \nabla + q\int_0^1 \lambda\mathbf{r}\times \mathbf{B}(\lambda\mathbf{r},t)\, d\lambda \right]^2 - q\int_0^1 \mathbf{r}\cdot\mathbf{E}(\lambda\mathbf{r},t)\, d\lambda,$$3

so that modes with $$H_{\rm LG} = \frac{1}{2M}\left[ - i \nabla + q\int_0^1 \lambda\mathbf{r}\times \mathbf{B}(\lambda\mathbf{r},t)\, d\lambda \right]^2 - q\int_0^1 \mathbf{r}\cdot\mathbf{E}(\lambda\mathbf{r},t)\, d\lambda,$$4 are suppressed. The Hamiltonian in an arbitrary gauge $$H_{\rm LG} = \frac{1}{2M}\left[ - i \nabla + q\int_0^1 \lambda\mathbf{r}\times \mathbf{B}(\lambda\mathbf{r},t)\, d\lambda \right]^2 - q\int_0^1 \mathbf{r}\cdot\mathbf{E}(\lambda\mathbf{r},t)\, d\lambda,$$5 is

$$H_{\rm LG} = \frac{1}{2M}\left[ - i \nabla + q\int_0^1 \lambda\mathbf{r}\times \mathbf{B}(\lambda\mathbf{r},t)\, d\lambda \right]^2 - q\int_0^1 \mathbf{r}\cdot\mathbf{E}(\lambda\mathbf{r},t)\, d\lambda,$$6

where the gauge is fixed by a Green-kernel decomposition $$H_{\rm LG} = \frac{1}{2M}\left[ - i \nabla + q\int_0^1 \lambda\mathbf{r}\times \mathbf{B}(\lambda\mathbf{r},t)\, d\lambda \right]^2 - q\int_0^1 \mathbf{r}\cdot\mathbf{E}(\lambda\mathbf{r},t)\, d\lambda,$$7, and the material polarization field is

$$H_{\rm LG} = \frac{1}{2M}\left[ - i \nabla + q\int_0^1 \lambda\mathbf{r}\times \mathbf{B}(\lambda\mathbf{r},t)\, d\lambda \right]^2 - q\int_0^1 \mathbf{r}\cdot\mathbf{E}(\lambda\mathbf{r},t)\, d\lambda,$$8

Different gauges remain related by the unitary transformation

$$H_{\rm LG} = \frac{1}{2M}\left[ - i \nabla + q\int_0^1 \lambda\mathbf{r}\times \mathbf{B}(\lambda\mathbf{r},t)\, d\lambda \right]^2 - q\int_0^1 \mathbf{r}\cdot\mathbf{E}(\lambda\mathbf{r},t)\, d\lambda,$$9

with $\psi_{\rm LG} = W^{-1}\psi_{\rm VG}$0 (Chivers-White et al., 7 Jun 2026).

For the Poincaré, or multipolar, choice,

$\psi_{\rm LG} = W^{-1}\psi_{\rm VG}$1

the regularised multipolar polarization is

$\psi_{\rm LG} = W^{-1}\psi_{\rm VG}$2

In the point-charge limit $\psi_{\rm LG} = W^{-1}\psi_{\rm VG}$3, this reduces to the standard multipolar polarization (Chivers-White et al., 7 Jun 2026).

A notable technical extension is the introduction of two independent cut-offs: $\psi_{\rm LG} = W^{-1}\psi_{\rm VG}$4 for the longitudinal Coulomb sector and $\psi_{\rm LG} = W^{-1}\psi_{\rm VG}$5 for the transverse multipolar sector. The “convolutional” multipolar transverse kernel is defined by

$\psi_{\rm LG} = W^{-1}\psi_{\rm VG}$6

so that the transverse polarization is effectively regularised by $\psi_{\rm LG} = W^{-1}\psi_{\rm VG}$7, while the Coulomb sector is regularised by $\psi_{\rm LG} = W^{-1}\psi_{\rm VG}$8 (Chivers-White et al., 7 Jun 2026). This separates regularisation of the longitudinal and transverse sectors without abandoning unitary equivalence.

The material energy

$\psi_{\rm LG} = W^{-1}\psi_{\rm VG}$9

contains self-energies and inter-atomic interactions. In Coulomb gauge, one recovers the regularised Coulomb potential

$\mathbf x\cdot\mathbf A(\mathbf x)=0.$0

In the regularised multipolar gauge with Lorentzian $\mathbf x\cdot\mathbf A(\mathbf x)=0.$1,

$\mathbf x\cdot\mathbf A(\mathbf x)=0.$2

and

$\mathbf x\cdot\mathbf A(\mathbf x)=0.$3

For large $\mathbf x\cdot\mathbf A(\mathbf x)=0.$4,

$\mathbf x\cdot\mathbf A(\mathbf x)=0.$5

while for small $\mathbf x\cdot\mathbf A(\mathbf x)=0.$6,

$\mathbf x\cdot\mathbf A(\mathbf x)=0.$7

The paper interprets this as a cut-off-dependent trade-off: small $\mathbf x\cdot\mathbf A(\mathbf x)=0.$8 makes individual multipolar terms weak but delocalises the polarization cloud, whereas large $\mathbf x\cdot\mathbf A(\mathbf x)=0.$9 localises material subsystems and suppresses direct inter-atomic interactions but makes the $(\mathbf A_{\mathrm T},\bm\Pi)$0 contribution non-negligible (Chivers-White et al., 7 Jun 2026).

For two atoms, the multipolar direct interaction is exponentially suppressed at separations $(\mathbf A_{\mathrm T},\bm\Pi)$1, because the polarization clouds overlap only weakly. In the unregularised multipolar gauge $(\mathbf A_{\mathrm T},\bm\Pi)$2, the direct interaction between well-separated atoms is strictly zero; with finite cut-off it re-emerges at short range (Chivers-White et al., 7 Jun 2026). This suggests that ultraviolet regularisation changes not the formal equivalence of Coulomb and multipolar gauges, but the practical balance between localization, perturbative bookkeeping, and short-range physics.

4. Periodic boundary conditions and many-body regularisation of electric multipoles

In periodic extended systems, ordinary position operators and their products are not well defined on the Hilbert space with periodic boundary conditions. The many-body operators

$(\mathbf A_{\mathrm T},\bm\Pi)$3

map periodic states خارج the periodic Hilbert space and can become non-normalisable in the thermodynamic limit. The regularisation adopted for multipole moments is therefore an exponentiation of the corresponding many-body operators (Wheeler et al., 2018).

For polarization, Resta’s operator is

$(\mathbf A_{\mathrm T},\bm\Pi)$4

and the polarization is extracted from the phase of its ground-state expectation value,

$(\mathbf A_{\mathrm T},\bm\Pi)$5

Wheeler, Wagner, and Hughes generalise this to higher multipoles. For the quadrupole moment,

$(\mathbf A_{\mathrm T},\bm\Pi)$6

and for a rectangular lattice,

$(\mathbf A_{\mathrm T},\bm\Pi)$7

Similarly, for the octupole moment,

$(\mathbf A_{\mathrm T},\bm\Pi)$8

(Wheeler et al., 2018).

These phases are defined modulo $(\mathbf A_{\mathrm T},\bm\Pi)$9, so the multipole moments are defined only modulo multipole quanta. A key structural condition is that the $\bm\Pi=-\mathbf D_{\mathrm T}$0-th multipole moment is uniquely defined only when all lower moments vanish modulo their respective quanta. Thus a bulk quadrupole requires vanishing polarization modulo the polarization quantum, and an octupole requires both vanishing polarization and vanishing quadrupole modulo their quanta (Wheeler et al., 2018).

The same work ties these operators to adiabatic response. For quadrupole order, the unitary generated by

$\bm\Pi=-\mathbf D_{\mathrm T}$1

couples to a dipole-current operator

$\bm\Pi=-\mathbf D_{\mathrm T}$2

and corresponds to adiabatically turning on a vector potential with uniform electric-field gradient,

$\bm\Pi=-\mathbf D_{\mathrm T}$3

Accordingly, the time derivative of the quadrupole moment is governed by a dipole current (Wheeler et al., 2018).

The many-body operators correctly identify topological quadrupole and octupole phases in tight-binding models, distinguish a bulk quadrupole moment from corner charges generated by edge polarization, and capture an adiabatic quadrupole pump (Wheeler et al., 2018). In this condensed-matter setting, regularised multipolar gauge refers not to a Hamiltonian gauge transformation between $\bm\Pi=-\mathbf D_{\mathrm T}$4 and $\bm\Pi=-\mathbf D_{\mathrm T}$5 couplings, but to a gauge-compatible regularisation of multipole observables under periodic boundary conditions.

5. Higher-form and lattice-regularised multipolar gauge in topological phases

A further generalisation appears in the field-theoretic description of multipole topological phases. In dipole-conserving systems, the relevant bulk response is reformulated in terms of electric higher-form symmetries. Gauging a 1-form electric symmetry introduces an antisymmetric 2-form gauge field $\bm\Pi=-\mathbf D_{\mathrm T}$6, with gauge transformations

$\bm\Pi=-\mathbf D_{\mathrm T}$7

and gauge-invariant combination $\bm\Pi=-\mathbf D_{\mathrm T}$8. The associated Maxwell-like Lagrangian is

$\bm\Pi=-\mathbf D_{\mathrm T}$9

(Dubinkin et al., 2020).

The decisive construction is a generalized 2-form Peierls substitution for ring-exchange processes. For a plaquette $(\mathbf A_{\mathrm T},-\mathbf E_{\mathrm T})$0,

$(\mathbf A_{\mathrm T},-\mathbf E_{\mathrm T})$1

and the ring-exchange operator acquires the phase $(\mathbf A_{\mathrm T},-\mathbf E_{\mathrm T})$2. In this formulation, the off-diagonal rank-2 gauge component $(\mathbf A_{\mathrm T},-\mathbf E_{\mathrm T})$3 is not treated as a second derivative of an ordinary vector potential, but as a compact plaquette holonomy of the 2-form field $(\mathbf A_{\mathrm T},-\mathbf E_{\mathrm T})$4 (Dubinkin et al., 2020). This regularises the multipolar gauge structure on a periodic lattice and avoids the obstruction that a uniform $(\mathbf A_{\mathrm T},-\mathbf E_{\mathrm T})$5 cannot generally be realised as a pure derivative of a globally defined $(\mathbf A_{\mathrm T},-\mathbf E_{\mathrm T})$6 on a torus.

The quadrupole response is then written as a topological term

$(\mathbf A_{\mathrm T},-\mathbf E_{\mathrm T})$7

where $(\mathbf A_{\mathrm T},-\mathbf E_{\mathrm T})$8 is identified as a Dixmier–Douady invariant. On a closed $(\mathbf A_{\mathrm T},-\mathbf E_{\mathrm T})$9-dimensional spacetime manifold,

$$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$00

so $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$01 is defined modulo $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$02 (Dubinkin et al., 2020). Mirror symmetries $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$03, $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$04, and $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$05 force $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$06, giving $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$07 or $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$08.

The same framework reinterprets the rank-2 Berry phase as a large 1-form gauge transformation of $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$09. Adiabatically changing a uniform $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$10 from $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$11 to $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$12 corresponds to inserting one unit of 2-form flux through the torus, and the Berry phase becomes

$$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$13

A higher-form Lieb–Schultz–Mattis theorem is also obtained: for a unique gapped symmetric ground state, the bulk polarization must satisfy

$$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$14

(Dubinkin et al., 2020).

In this topological setting, regularised multipolar gauge means that multipole gauge fields are encoded as compact higher-form holonomies on lattice cells, rather than as singular continuum tensor potentials. This suggests a broad conceptual continuity with the many-body operator approach: in both cases, compactification and quantized phase data replace ill-defined unbounded multipole coordinates.

6. Conceptual synthesis, misconceptions, and scope

Several misconceptions are explicitly addressed by the cited literature. First, multipolar gauge is not a theory fundamentally inequivalent to Coulomb gauge. In constrained nrQED, Poincaré gauge and multipolar QED are identical, and both are unitarily equivalent to Coulomb-gauge QED; the earlier controversy arose from using “canonical momentum” for two different objects (Stokes et al., 2021). Second, beyond the electric dipole approximation, one cannot simply add magnetic dipole or electric quadrupole terms by hand to a heuristic length-gauge Hamiltonian and expect gauge consistency. The finite-order gauge partner of $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$15 is uniquely $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$16, and the nonlinear $$\mathbf{E}(\mathbf{x},t) = -\dot{\mathbf{A}}(\mathbf{x},t)$$17-type and quadratic magnetic terms cannot be omitted if strict gauge equivalence is required (Anzaki et al., 2018). Third, in regularised nrQED, a multipolar gauge with finite cut-off does not eliminate direct inter-atomic interactions exactly; it suppresses them exponentially when polarization clouds have negligible overlap, while introducing a cut-off-dependent trade-off between subsystem localisation and the size of individual interaction terms (Chivers-White et al., 7 Jun 2026).

The literature also shows that “regularisation” is context dependent. In strong-field atomic physics, it is an order-by-order truncation preserving gauge equivalence between minimal-coupling and multipolar descriptions (Anzaki et al., 2018). In nonrelativistic QED with ultraviolet control, it is a smearing of charge and polarization densities that keeps Coulomb and multipolar gauges unitarily related while rendering self-energies finite (Chivers-White et al., 7 Jun 2026). In periodic crystals, it is the replacement of ill-defined many-body multipole operators by exponentiated unitaries compatible with periodic boundary conditions (Wheeler et al., 2018). In higher-order topological phases, it is the replacement of continuum rank-2 potentials by compact higher-form lattice holonomies and topological response terms (Dubinkin et al., 2020).

A plausible unifying implication is that a regularised multipolar gauge is best understood not as a single formula, but as a disciplined procedure for preserving the multipolar description under whatever obstruction is present: finite-order truncation, ultraviolet divergence, periodicity, or compact lattice topology. In all cases surveyed here, the central criterion is the same: the multipolar formulation must remain connected to an underlying gauge structure by an explicit and controlled construction.