Podolsky Quantum Electrodynamics (POQED)

• Podolsky QED is a higher-derivative extension of standard QED that introduces quadratic derivative terms into the Maxwell Lagrangian, establishing an intrinsic energy scale.
• It splits the photon sector into massless and massive modes, which regularizes ultraviolet divergences and yields finite self-energy for point charges.
• The framework has significant implications in plasma physics, black hole thermodynamics, and condensed matter, using advanced quantization and thermal methods.

Podolsky Quantum Electrodynamics (POQED) is a higher-derivative generalization of quantum electrodynamics that augments the Maxwell action by introducing quadratic terms in the derivatives of the field-strength tensor, thereby furnishing the theory with an intrinsic length or energy scale and profoundly altering its behavior at both ultraviolet and infrared limits. This extension provides a natural, gauge-invariant regularization mechanism that resolves or softens a variety of divergences encountered in both the classical (point-charge) and quantum domains, with ramifications for field propagation, self-force phenomena, vacuum polarization, plasma physics, black hole thermodynamics, and condensed matter applications.

1. Fundamental Formulation and Quantization

The core Lagrangian of Podolsky electrodynamics is

$$\mathcal{L}_{\text{Pod}} = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu} + \frac{a^2}{2}\partial_\lambda F_{\mu\nu}\partial^\lambda F^{\mu\nu}$$

where $a$ (or $a = 1/m_p$) is the Podolsky length parameter, and the higher-derivative operator raises the field equations to fourth order. The quantum theory preserves Lorentz and gauge invariance and splits the photon sector into massless and massive modes:

• The massless sector ($p^2 = 0$) retains the standard Maxwell behavior.
• The massive sector ($p^2 = m_p^2$, $m_p = 1/a$) yields a Proca-like spin-1 mode.

Canonical and path-integral quantizations of the original higher-derivative model reveal a constraint structure consisting of both first-class (gauge) and second-class constraints. A reduced-order formulation, achieved by introducing an auxiliary massive vector field $B_\mu$, recasts the fourth-order equations into a coupled system of second-order equations (Thibes, 2016). The effective quantum theory is then constructed using Senjanovic’s procedure for second-class constraints and the Batalin-Fradkin-Vilkovisky (BFV) approach for the first-class sector, yielding a BRST-invariant path integral with the final action (in configuration space) manifestly covariant.

In the Matsubara-Fradkin thermal formalism, the partition function is established from the full density matrix, with gauge invariance fixed at the Lagrangian level using the Nakanishi method. The structure of the quantum theory at finite temperature emerges naturally via path integral methods, periodic/antiperiodic boundary conditions, and the inclusion of auxiliary and ghost fields (Bonin et al., 2011). This ensures the validity of the Dyson-Schwinger-Fradkin equations, yielding non-perturbative relationships between full thermal propagators and vertices; Ward-Fradkin-Takahashi (WFT) identities that enforce transversality of polarization tensors even at finite temperature, and the explicit form of the polarization tensor in a medium, where both the Podolsky mass parameter and the statistical variables (through the rest-frame $u^\mu$) appear as essential ingredients.

2. Propagation, Green Functions, and Classical Regularization

The operator structure of POQED, $(1 + a^2\square)\square$, leads to a propagator: $$D_{\mu\nu}(k) = -i\frac{1}{k^2(1 - a^2 k^2)}\left[ \eta_{\mu\nu} - \ldots \right]$$ which, in coordinate space, yields the Green function

$$G^{\text{Pod}}(x) = G^{\Box}(x) - a^2 G^{\text{KG}}(x)$$

with $G^{\Box}$ (Maxwell) and $G^{\text{KG}}$ (Klein-Gordon) components (Lazar, 2020). This structure induces a double-Yukawa profile in the interparticle potential: $$V(r) \sim \frac{e^{-m_D r}}{r} - \frac{e^{-M r}}{r}$$ with $m_D$ the Debye mass and $M = 1/a$ the Podolsky mass scale (Singh et al., 3 Aug 2025). The short-distance (ultraviolet) singularity of the Coulomb law is regularized by the heavy, massive pole, resulting in a finite self-energy for point charges. For instance, in the context of self-force and motion, the fourth-order field equations support Green function techniques that render the self-field—and hence self-force—finite for all worldlines “bounded away from the backward light-cone,” even as the motive point charge crosses singular hypersurfaces, with directional discontinuities tamed by suitable averaging (Gratus et al., 2015).

This regularization persists even in curved backgrounds with topological defects: the self-interaction energy of a stationary charge in conical (cosmic string) and monopole spacetimes is finite everywhere, directly computable from the regularized potential

$$A_0(r) = \frac{q}{|r - r_c|}\left(1 - e^{-\mu|r - r_c|}\right)$$

and involves integrals over well-behaved Green functions, with the angular defect parameter governing qualitative behavior, in contrast to the divergent case for Maxwell theory (Zayats, 2016).

3. Vacuum Polarization, Renormalization, and Unitarity

At the quantum level, the presence of the higher-derivative operator directly modifies the one-loop polarization tensor and the photon propagator structure. In momentum space, the photon propagator assumes the subtraction form

$$D_{\mu\nu}(p) \propto \eta_{\mu\nu}\left( \frac{1}{p^2} - \frac{1}{p^2 + m_p^2} \right)$$

thereby introducing a massless and a massive pole (Montenegro, 2022). This ensures that two- and three-point correlation functions are rendered ultraviolet finite, while the vacuum polarization remains logarithmically divergent at order $e^2$, as in standard QED.

The polarization tensor, derived by Källén’s method in the Heisenberg picture, maintains gauge invariance explicitly: $$K^{\mu\nu}(p) = \left[a^2p^2\,p^\mu p^\nu - a^2p^4\,g^{\mu\nu} + p^2\,g^{\mu\nu} - p^\mu p^\nu\right]\,G(p^2)$$ where $p_\mu K^{\mu\nu} = 0$; charge renormalization is properly enforced by the subtraction $G(p^2) - G(0)$, yielding well-defined observable currents and a finite, renormalized coupling (Montenegro, 2022). This subtraction method preserves both unitarity and stability, circumventing Ostrogradski ghosts by proper gauge fixing and interpretation of the higher-derivative term as kinetic rather than interaction.

Precision QED tests, such as the anomalous magnetic moment of the electron, are used to set stringent bounds on the Podolsky parameter: $m_p \geq 3.7595\times 10^{10}~\text{eV}$ indicating that deviations from QED would only manifest in high-energy experiments (Montenegro, 2022). In planar (2+1)D reductions, the anomaly vanishes at one-loop, and the structure of radiative corrections is modified due to kinematical peculiarities (Montenegro et al., 2022).

4. Finite-Temperature POQED and Collective Phenomena

POQED at finite temperature (and finite density) is systematically quantized using the Matsubara-Fradkin formalism and, for scattering processes, the real-time Thermo Field Dynamics (TFD) technique (Bonin et al., 2011, Cabral et al., 8 Sep 2025). The path integral representation naturally incorporates periodicity (bosonic) or anti-periodicity (fermionic) in Euclidean time. This leads to:

• Explicit Dyson-Schwinger-Fradkin equations relating fully dressed propagators and vertex functions in equilibrium.
• Ward-Fradkin-Takahashi identities guaranteeing gauge invariance and transversality of the polarization tensor even at nonzero temperature.

The polarization tensor in a heat bath acquires an extended tensor structure: $$\Pi_{\mu\nu}(k, \omega) = A(k, \omega)\left(\delta_{\mu\nu} - \frac{k_\mu k_\nu}{k_B^2}\right) + B(k, \omega)...$$ where scalars $A, B$ depend on the Podolsky mass and medium variables, and the rest-frame $u^\mu$ induces anisotropy (e.g., screening and plasma oscillations).

In plasma applications, the higher-derivative operator modifies the photon dispersion relation: $(1 + a^2 k^2)\omega^2 = c^2 k^2$ introducing two distinct propagating modes, with a high-momentum regularization naturally present. The effective photon-plasma interaction Hamiltonian contains new mode-mode coupling terms, and tracing out photonic degrees of freedom produces non-local kernels (in imaginary time), signifying non-Markovian behavior in finite systems (Singh et al., 3 Apr 2025). Thermodynamic quantities, such as the Debye mass and electrical conductivity, are corrected by permille-level shifts at two-loop order, with the double-Yukawa screened potential removing the Coulomb singularity at small separations (Singh et al., 3 Aug 2025). Gauge symmetry enforces vanishing magnetic screening mass at zero momentum for all perturbative orders.

5. Generalizations, Extensions, and External Structures

The POQED framework is flexible and adaptable to further generalization. For example:

• Lorentz-violating extensions: Addition of Carroll-Field-Jackiw (CFJ) terms to POQED gives rise to an electromagnetic vacuum with birefringence, dichroism, and frequency-dependent refractive indices, as well as photon–massive-mode oscillations. The presence of higher-derivative terms in a Lorentz non-invariant background modifies the photon dispersion relations, induces mass, and leads to Cherenkov-like radiation even in vacuum (Ferreira et al., 13 Sep 2024).
• Gravity and Black Hole Physics: Coupling BP electrodynamics to gravity in (2+1)D and (3+1)D has yielded both regular black hole solutions and exotic wormholes. BP corrections affect the horizon structure, Hawking temperature (which is suppressed by GUP corrections), and may induce modifications to observable signatures such as black hole shadows or Keplerian frequencies (Frizo et al., 2022, Ahmed et al., 15 Jul 2025, Moreira et al., 16 Jul 2025).
• Dimensional Reduction and Condensed Matter: POQED leads, upon dimensional reduction (with planar matter current but bulk gauge field), to “Pseudo Generalized QED” (PGQED) in (2+1)D, retaining a non-local effective gauge sector that encodes the ultraviolet regularization (Junior et al., 21 Oct 2025). In the strong-coupling regime, PGQED supports dynamical chiral symmetry breaking and gap opening in the Dirac spectrum, relevant for the understanding of screening and collective effects in graphene and other Dirac materials.

6. Regularization, Historical Parallels, and Mathematical Rigor

The Podolsky operator provides a precise, Lorentz-invariant, and gauge-invariant regularization scheme akin to Feynman’s relativistic cut-off. Feynman’s procedure of replacing the massless photon propagator with a difference of massless and massive propagators is formally equivalent to the Hamiltonian approach of Podolsky, as clarified by direct comparison of their respective field equations and the identification $a = 1/\lambda_0$ (Chapagain, 18 Nov 2024). This equivalence extends to self-energy calculations for particles of arbitrary spin and ensures Lorentz-invariant mass renormalizations.

The rigorous mathematical treatment extends to quantum field theory on the Poincaré group, where fields are defined on homogeneous spaces $\mathbb{R}^{1,3} \times S^2$, yielding naturally convergent S-matrix elements due to the extended internal structure; this group-theoretical approach is in close harmony with the renormalization mechanisms of POQED (Varlamov, 2011).

7. Summary Table: Core Features of POQED

Aspect	Standard QED	POQED Variant
Lagrangian Derivative Order	Second	Includes fourth-order (higher derivative)
Gauge Invariance	Preserved	Preserved (with careful gauge-fixing)
Photon Spectrum	Massless only	Massless plus massive (Proca-like)
Ultraviolet Behavior	Divergent (counterterms)	Regularized, improved convergence
Self-force for Point Charge	Infinite (singular)	Finite (regularized by massive mode)
Vacuum Polarization	Transverse, divergent	Transverse, divergence structure tamed
Finite Temperature Structure	Plasma modes, screening	Additional screening/polarization, double-Yukawa
Applicability	EM, HEP, condensed matter	Regularization, plasma, gravity, condensed matter

8. Outlook and Implications

Podolsky Quantum Electrodynamics provides a mathematically consistent, physically robust generalization of QED with built-in regularization for both classical and quantum divergences. The formalism ensures unitarity, stability, and renormalizability while opening avenues for finite self-interaction, improved thermodynamic and transport calculations in plasmas, regularized black hole solutions, Lorentz-violating phenomenology (such as refractive vacuum birefringence), and condensed matter realizations with non-local interaction potentials.

These structural advances, encapsulated in the modified kinetic operator and propagator, enable the systematic paper of strong-field, high-temperature, and strongly coupled regimes otherwise inaccessible to standard QED techniques, and provide testable predictions that can be constrained by precision experiments, astrophysical observations, and condensed matter platforms. Future directions include lattice simulations of massive gauge modes, collider searches for deviations at high energy, and further explorations of the interplay between higher-derivative dynamics, topology, and quantum gravity effects.