Bopp–Podolsky Electrodynamics Overview

• Bopp–Podolsky electrodynamics is a linear, Lorentz- and gauge-invariant higher-derivative extension of Maxwell theory that regularizes the Coulomb singularity.
• It decomposes the electromagnetic field into a massless Maxwell sector and a massive Proca-like sector, leading to finite self-force and retarded potentials.
• The theory has broad applications, from addressing point-charge self-interaction to shaping curved-spacetime solutions and nonlinear PDE models.

Searching arXiv for recent and foundational papers on Bopp–Podolsky electrodynamics to support the article. arxiv_search(query="Bopp-Podolsky electrodynamics", max_results=10, sort_by="relevance") arXiv search results were consulted for "Bopp-Podolsky electrodynamics", including foundational and recent work on self-force, Green functions, curved-spacetime generalizations, nonlinear Schrödinger/Dirac couplings, and black-hole applications. Bopp–Podolsky electrodynamics is a linear, Lorentz-invariant, gauge-invariant higher-derivative generalization of Maxwell theory in which the electromagnetic sector is modified by a single short-distance scale. In the literature it appears both as Bopp–Podolsky (BP) and, in the fuller historical attribution, as Bopp–Landé–Thomas–Podolsky (BLTP) electrodynamics. Its central purpose is to soften the ultraviolet singularities of the Maxwell field of a point charge without abandoning linearity or $U(1)$ gauge symmetry. The resulting theory has fourth-order field equations, admits a decomposition into a massless Maxwell sector and a massive Proca-like sector, replaces the Coulomb singularity by a finite static potential, and converts the classical self-force problem into a finite but history-dependent interaction (Gratus et al., 2015).

1. Origins and defining field equations

The defining modification is the addition of a term quadratic in derivatives of the field strength. In one standard flat-space convention,

$$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$

with $F_{ab}=\partial_aA_b-\partial_bA_a$. In Lorenz gauge this yields

$\square A_a-\ell^2\square^2 A_a=-4\pi j_a,$

or equivalently

$$\partial^bF_{ba}-\ell^2\square\,\partial^bF_{ba}=-4\pi j_a.$$

In BLTP notation the same structural modification is written as

$$\left(I-\kappa^{-2}\Box\right)\partial^cF_{cd}=-4\pi j_d,\qquad \partial_aF_{bc}+\partial_bF_{ca}+\partial_cF_{ab}=0,$$

with $\kappa>0$ Bopp’s wave number (Gratus et al., 2015).

Different authors parameterize the new scale by $\ell$, $a$, $\kappa^{-1}$, or $$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$0. The notational differences do not alter the common structural fact that BP electrodynamics is a fourth-order but linear gauge theory. In vacuum, one representative form is

$$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$1

which displays directly that the Podolsky operator multiplies the Maxwell operator. The curved-spacetime literature extends the same idea by replacing partial derivatives with covariant derivatives and, in more general versions, including explicit curvature couplings proportional to $$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$2 (Zayats, 2013).

A frequent misconception is that the extra massive mode is introduced by breaking gauge symmetry in a Proca-like manner. The BP construction does not do this. The higher-derivative term is built from $$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$3, so the model remains $$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$4-gauge invariant while still generating a massive propagation scale in its spectrum (Cuzinatto et al., 2016).

2. Mode decomposition, propagators, and Green functions

A basic structural result is the decomposition into a Maxwell field and a massive vector field. One formulation writes

$$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$5

where

$$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$6

and

$$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$7

A closely related convention uses

$$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$8

so that the second term behaves as a massive Proca-like mode with mass scale $$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$9 (Gratus et al., 2015).

At the propagator level, the characteristic denominator is

$F_{ab}=\partial_aA_b-\partial_bA_a$0

This exact algebraic split underlies the interpretation of BP electrodynamics as containing an intrinsic Pauli–Villars-like regularization: the photon propagator is the difference between a massless pole and a massive pole. In Lorenz gauge one obtains

$F_{ab}=\partial_aA_b-\partial_bA_a$1

The parameter $F_{ab}=\partial_aA_b-\partial_bA_a$2 is therefore both the higher-derivative length scale and the inverse regulator mass scale $F_{ab}=\partial_aA_b-\partial_bA_a$3 (Ji et al., 2019).

The same structure appears in classical statics. For a point charge, the electrostatic potential becomes

$F_{ab}=\partial_aA_b-\partial_bA_a$4

instead of the singular Coulomb potential $F_{ab}=\partial_aA_b-\partial_bA_a$5. In another notation,

$F_{ab}=\partial_aA_b-\partial_bA_a$6

and the electrostatic field energy is finite: $F_{ab}=\partial_aA_b-\partial_bA_a$7 The same finite potential can also be realized in ordinary electrostatics by replacing a point source with the smeared density

$F_{ab}=\partial_aA_b-\partial_bA_a$8

which yields

$F_{ab}=\partial_aA_b-\partial_bA_a$9

This makes the short-distance regularization geometrically transparent: BP theory makes a point charge behave, at the level of its static potential, like a finite-size charge distribution (Zayats, 2013).

Retarded Green functions show the same regularizing mechanism in spacetime. The BP Green kernel can be written as a Maxwell term minus a Klein–Gordon term, $\square A_a-\ell^2\square^2 A_a=-4\pi j_a,$0, and in $\square A_a-\ell^2\square^2 A_a=-4\pi j_a,$1 dimensions it is supported inside the forward light cone rather than purely on the light cone. The resulting retarded fields exhibit decreasing oscillations inside the cone, and the retarded potentials depend on the entire source history up to the retarded time. This “tail” or “afterglow” behavior is the spacetime counterpart of the softened static singularity (Lazar, 2020).

3. Point charges, self-force, and radiation reaction

The most developed classical application of BP electrodynamics is the point-charge self-force problem. In Maxwell–Lorentz theory the field diverges on the worldline, the self-force is formally infinite, and mass renormalization leads to the Abraham–Lorentz–Dirac equation with its familiar runaway and pre-acceleration pathologies. BP theory changes the problem qualitatively because the point-charge field is no longer singular in the same way. For a broad class of timelike worldlines, the self-force becomes finite and can be written as a convergent history integral (Gratus et al., 2015).

For a particle on a timelike worldline $\square A_a-\ell^2\square^2 A_a=-4\pi j_a,$2, the BLTP retarded potential is an integral over past proper time with Bessel kernel $\square A_a-\ell^2\square^2 A_a=-4\pi j_a,$3, and the self-field entering the equation of motion is

$\square A_a-\ell^2\square^2 A_a=-4\pi j_a,$4

where $\square A_a-\ell^2\square^2 A_a=-4\pi j_a,$5. For $\square A_a-\ell^2\square^2 A_a=-4\pi j_a,$6 non-intersecting worldlines the resulting equation of motion is

$\square A_a-\ell^2\square^2 A_a=-4\pi j_a,$7

This formula is not introduced as an ad hoc correction. One derivation starts from distributional conservation of total four-momentum, and another derives the same equation from a variational principle formulated directly with retarded fields. The self-force coincides with earlier proposals by Zayats and by Gratus–Perlick–Tucker, but here it emerges from energy-momentum conservation alone (Baza et al., 2019).

The earlier self-force analysis establishes precise convergence criteria. A worldline “bounded away from the past light-cone” of an event has absolutely convergent potential, field, and self-force integrals. Even when the BP field is bounded, it has a directional discontinuity as one approaches the particle. The finite value of the field on the worldline is then assigned by a Lorentz-covariant averaging over shrinking spheres, which removes the directional singular part and yields a finite Lorentz self-force. The resulting equation of motion is an integro-differential equation with genuine memory of the entire past trajectory (Gratus et al., 2015).

BP theory does not suppress radiation reaction; it regularizes it. In the small-$\square A_a-\ell^2\square^2 A_a=-4\pi j_a,$8 limit, the self-force expansion contains a leading term

$\square A_a-\ell^2\square^2 A_a=-4\pi j_a,$9

which is absorbed into the mass, and a finite remainder equal to the Lorentz–Dirac radiation-reaction term

$$\partial^bF_{ba}-\ell^2\square\,\partial^bF_{ba}=-4\pi j_a.$$0

Thus the theory reproduces the Abraham–Lorentz–Dirac structure in the appropriate singular limit while remaining finite at nonzero BP scale (Gratus et al., 2015).

Special trajectories reveal additional structure. For uniformly accelerated straight-line motion the self-force remains parallel to the acceleration and can be written

$$\partial^bF_{ba}-\ell^2\square\,\partial^bF_{ba}=-4\pi j_a.$$1

with

$$\partial^bF_{ba}-\ell^2\square\,\partial^bF_{ba}=-4\pi j_a.$$2

The electromagnetic contribution to inertia is therefore acceleration-dependent and decreases as the proper acceleration $$\partial^bF_{ba}-\ell^2\square\,\partial^bF_{ba}=-4\pi j_a.$$3 increases, approaching the bare mass $$\partial^bF_{ba}-\ell^2\square\,\partial^bF_{ba}=-4\pi j_a.$$4 at very large acceleration. This paper interprets the result as a possible route, in principle, to separate bare and electromagnetic mass experimentally (Zayats, 2013).

A more recent test problem considers straight-line motion driven by a constant applied electric field in a BLTP vacuum. There the exact self-force defines a Volterra integral equation for the acceleration. The small-$$\partial^bF_{ba}-\ell^2\square\,\partial^bF_{ba}=-4\pi j_a.$$5 expansion has vanishing $$\partial^bF_{ba}-\ell^2\square\,\partial^bF_{ba}=-4\pi j_a.$$6 and $$\partial^bF_{ba}-\ell^2\square\,\partial^bF_{ba}=-4\pi j_a.$$7 terms; the first nonzero contribution is the previously known $$\partial^bF_{ba}-\ell^2\square\,\partial^bF_{ba}=-4\pi j_a.$$8 term

$$\partial^bF_{ba}-\ell^2\square\,\partial^bF_{ba}=-4\pi j_a.$$9

and the new $$\left(I-\kappa^{-2}\Box\right)\partial^cF_{cd}=-4\pi j_d,\qquad \partial_aF_{bc}+\partial_bF_{ca}+\partial_cF_{ab}=0,$$0 correction changes the long-time behavior substantially. The main conclusion is that the $$\left(I-\kappa^{-2}\Box\right)\partial^cF_{cd}=-4\pi j_d,\qquad \partial_aF_{bc}+\partial_bF_{ca}+\partial_cF_{ab}=0,$$1-truncated motion is reliable only for short times satisfying $$\left(I-\kappa^{-2}\Box\right)\partial^cF_{cd}=-4\pi j_d,\qquad \partial_aF_{bc}+\partial_bF_{ca}+\partial_cF_{ab}=0,$$2, and that its long-time periodic or runaway-looking behavior should not be interpreted as a pathology of the full BLTP theory (McGuigan et al., 10 Jun 2025).

4. Curved spacetime, self-interaction, and black-hole exteriors

The curved-spacetime generalization retains the higher-derivative Maxwell term and may also include explicit curvature couplings. One representative matter Lagrangian is

$$\left(I-\kappa^{-2}\Box\right)\partial^cF_{cd}=-4\pi j_d,\qquad \partial_aF_{bc}+\partial_bF_{ca}+\partial_cF_{ab}=0,$$3

This extension is again linear in the gauge field, but the field equations and stress tensor acquire higher-derivative and curvature-coupling pieces (Cuzinatto et al., 2017).

In static, spherically symmetric black-hole exteriors, the BP field equation splits into a non-homogeneous asymptotically massless sector and a homogeneous asymptotically massive sector. Using Bekenstein’s integral method, the homogeneous sector is shown to vanish outside the event horizon. In the simplest $$\left(I-\kappa^{-2}\Box\right)\partial^cF_{cd}=-4\pi j_d,\qquad \partial_aF_{bc}+\partial_bF_{ca}+\partial_cF_{ab}=0,$$4 case, the surviving non-homogeneous field is exactly Maxwell-like, so the exterior geometry is Reissner–Nordström. The same analysis further concludes that the only exterior solution consistent with the weak and null energy conditions is the Maxwell one; non-Maxwellian exterior configurations are energetically excluded in that setting (Cuzinatto et al., 2017).

The self-interaction problem can also be transported to curved backgrounds with topological defects. For a static point charge in the straight cosmic string spacetime,

$$\left(I-\kappa^{-2}\Box\right)\partial^cF_{cd}=-4\pi j_d,\qquad \partial_aF_{bc}+\partial_bF_{ca}+\partial_cF_{ab}=0,$$5

while for a global monopole,

$$\left(I-\kappa^{-2}\Box\right)\partial^cF_{cd}=-4\pi j_d,\qquad \partial_aF_{bc}+\partial_bF_{ca}+\partial_cF_{ab}=0,$$6

In both cases the BP self-energy is finite everywhere, including at the defect or singularity, and the usual renormalization procedure is not required. The sign of the angular defect controls whether the self-force is attractive or repulsive: for $$\left(I-\kappa^{-2}\Box\right)\partial^cF_{cd}=-4\pi j_d,\qquad \partial_aF_{bc}+\partial_bF_{ca}+\partial_cF_{ab}=0,$$7 the charge is attracted toward the defect, and for $$\left(I-\kappa^{-2}\Box\right)\partial^cF_{cd}=-4\pi j_d,\qquad \partial_aF_{bc}+\partial_bF_{ca}+\partial_cF_{ab}=0,$$8 it is repelled (Zayats, 2016).

These curved-spacetime results also sharpen a recurring point of interpretation. BP electrodynamics regularizes the short-distance behavior of the electromagnetic field, but it does not guarantee that every gravitational sector sourced by the field is physically admissible. Energy conditions, horizon regularity, and asymptotic assumptions remain decisive in selecting which BP modes can persist in a given background (Cuzinatto et al., 2017).

5. Lower-dimensional compact objects, BTZ-type solutions, and wormhole claims

In $$\left(I-\kappa^{-2}\Box\right)\partial^cF_{cd}=-4\pi j_d,\qquad \partial_aF_{bc}+\partial_bF_{ca}+\partial_cF_{ab}=0,$$9 dimensions, BP electrodynamics has been used as the matter source for charged BTZ-like geometries. With

$\kappa>0$0

the perturbative analysis around a BTZ background produces first- and second-order corrections controlled mainly by the parameter $\kappa>0$1. The corrected electric field acquires short-distance terms such as $\kappa>0$2 and $\kappa>0$3, while the metric functions pick up inverse-power and logarithmic deformations. These corrections are strongest near the origin and decay asymptotically, so the spacetime remains BTZ-like far from the center (Maluf et al., 25 Feb 2025).

Within the validity regime of that perturbative treatment, no wormhole solution is found. The near-horizon and inner structure remain black-hole-like, and the analysis explicitly reports no evidence for wormholes or other non-black-hole compact objects in the $\kappa>0$4-dimensional theory. Energy-condition violations can occur when $\kappa>0$5, but they are described as limited and insufficient to justify topology change in the controlled perturbative region (Maluf et al., 25 Feb 2025).

Thermodynamic studies of the same class of $\kappa>0$6-dimensional BP-charged BTZ-type black holes compute the Hawking temperature via the Hamilton–Jacobi tunneling method,

$\kappa>0$7

and show explicit dependence on the cosmological constant and BP coupling. First- and second-order corrections modify the electric field, the horizon structure, entropy, and heat capacity. The reported behavior includes marginal violation of energy conditions in some parameter ranges, but the thermodynamic description remains well-defined (Moreira et al., 16 Jul 2025).

A related study of geodesics, scalar fields, and GUP-corrected thermodynamics uses the same charged BTZ-like background and analyzes null geodesics, effective potentials, shadow radii, scalar-wave propagation, and Keplerian frequencies. In that treatment the BP coupling $\kappa>0$8 produces explicit $\kappa>0$9 metric corrections and a $\ell$0 correction to the orbital frequency, thereby shifting the circular-photon-orbit and thermodynamic structure relative to the standard charged BTZ case (Ahmed et al., 15 Jul 2025).

The wormhole literature is more heterogeneous. One four-dimensional construction starts from a static BP-supported wormhole, rewrites it in Morris–Thorne form, and then generates a rotating counterpart whose shadows can be smooth or cuspy depending on parameter choices; the cusp arises from the coexistence of outer unstable photon orbits and throat-supported unstable orbits (Raza et al., 2024). The lower-dimensional compact-object analysis, by contrast, reports no evidence of wormholes in its perturbative domain (Maluf et al., 25 Feb 2025). This suggests that wormhole interpretations are model-dependent and sensitive to dimension, ansatz, and approximation scheme.

6. Matter-coupled nonlinear systems and reduced-order formulations

The electrostatic sector of BP electrodynamics supplies a regularized nonlocal kernel that has become standard in several nonlinear PDE models: $\ell$1 In the Schrödinger–Bopp–Podolsky system, standing waves satisfy

$\ell$2

with constrained or unconstrained variational formulations depending on the regime. For the electrostatic model in $\ell$3, nontrivial solutions exist for small coupling $\ell$4 when $\ell$5, and for arbitrary $\ell$6 when $\ell$7. In the radial case, solutions converge to those of the classical Schrödinger–Poisson system as $\ell$8 (d'Avenia et al., 2018).

The normalized-ground-state problem for the mass-supercritical Schrödinger–Bopp–Podolsky equation uses the constrained energy

$\ell$9

on the $a$0-sphere $a$1. In the regime $a$2, there exists $a$3 such that for every $a$4 the stationary problem has a positive weak solution at the mountain-pass level. The same work proves asymptotic concentration toward the classical NLS ground state $a$5, establishes radial symmetry and uniqueness up to translation in several parameter ranges, and shows strong instability of the associated standing waves at the mountain-pass level (Huang et al., 2024).

The Dirac–Bopp–Podolsky system replaces the Poisson equation by

$a$6

inside a nonlinear Dirac equation. The resulting nonlocal quartic term

$a$7

is treated variationally in a strongly indefinite setting. Under periodicity and growth hypotheses on the potential and nonlinearity, at least one nontrivial weak solution pair $a$8 exists, obtained via minimax arguments and Cerami sequences (Missaoui, 2023).

At the field-theoretic level, BP electrodynamics can be reformulated in reduced order by introducing an auxiliary vector field $a$9. One representative reduced-order Lagrangian is

$\kappa^{-1}$0

Eliminating $\kappa^{-1}$1 recovers the original BP field equation

$\kappa^{-1}$2

This split makes the massless and massive sectors explicit and simplifies canonical analysis on the null plane; after elimination of second-class constraints, the reduced-order model becomes a consistent Abelian first-class theory (Bertin et al., 2024).

The same reduced-order logic is developed for scalar BP electrodynamics coupled to a charged scalar field. There one obtains propagators in higher-order linear covariant, light-front, and doubly transverse light-front gauges, as well as a symmetric gauge-invariant improved energy-momentum tensor for both the original higher-derivative and reduced-order formulations. The photon sector again exhibits a massless pole and a massive pole at $\kappa^{-1}$3, now embedded in a full interacting matter theory (Oliveira et al., 2020).

7. Cosmology, finite temperature, and current interpretive status

The cosmological literature treats the BP photon content as a thermal gas containing both massless and massive modes. The dispersion relations are

$\kappa^{-1}$4

Using the corresponding partition function, one obtains an equation of state $\kappa^{-1}$5 that deviates from the pure-radiation value $\kappa^{-1}$6 but only modestly. The minimum quoted value of the barotropic parameter is

$\kappa^{-1}$7

Even with the massive BP photon mode included, the Friedmann evolution remains essentially radiation-like, with the scale factor still behaving as

$\kappa^{-1}$8

to good approximation (Cuzinatto et al., 2016).

A more recent finite-temperature QED treatment studies electron–proton plasmas through one and two loops with dimensional regularization and hard-thermal-loop resummation. In that framework the higher-derivative Podolsky operator introduces no new ultraviolet divergences; all counterterms reduce to the usual photon wave-function renormalization $\kappa^{-1}$9. The static inter-particle force becomes a double-Yukawa potential,

$$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$00

where $$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$01. The first term is the ordinary Debye piece, and the second is the opposite-signed Podolsky contribution that removes the Coulomb singularity at short distance (Singh et al., 3 Aug 2025).

The same thermal analysis shows that gauge symmetry forces the transverse photon self-energy to vanish at zero momentum, so no magnetic screening mass appears perturbatively. The dc electrical conductivity exceeds its QED value only by a tiny amount: $$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$02 for representative parameters $$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$03 and $$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$04. Conditions for observable Podolsky plasmons and cosmological bounds on $$S[A] = \int \left(\frac{1}{16\pi}F^{ab}F_{ab}+\frac{\ell^2}{16\pi}\partial^c F^{ab}\partial_c F_{ab}-A_a j^a\right)\, d^4x,$$05 are also identified, making the BP scale a thermodynamic and plasma-physics parameter rather than only a classical regularizer (Singh et al., 3 Aug 2025).

The interpretive status of the massive BP mode remains nuanced. Propagator analyses emphasize its Pauli–Villars-like role (Ji et al., 2019), the cosmological discussion notes the usual ghost concerns for higher-derivative theories (Cuzinatto et al., 2016), and the finite-temperature BRST treatment places the Podolsky pole outside the physical Hilbert space in a Lee–Wick/BRST sense (Singh et al., 3 Aug 2025). What is not in dispute across these distinct approaches is the core structural content: BP electrodynamics preserves gauge invariance, replaces the Coulomb singularity by a finite short-distance profile, and converts the self-interaction of a point charge from a divergent local problem into a controlled nonlocal one.