Nonlocal Quantum Electrodynamics

• Nonlocal QED is a quantum field theory extension where electromagnetic interactions are mediated by entire function smearing over multiple spacetime points while maintaining gauge invariance.
• It achieves ultraviolet finiteness and improved renormalization by damping high-momentum contributions, addressing anomalies such as the lepton g-2 and modified Coulomb potentials.
• The framework modifies Feynman rules and quantization procedures with gauge-link corrections, offering new insights for atomic physics, high-energy scattering, and condensed matter applications.

Nonlocal Quantum Electrodynamics (QED) denotes a class of quantum field theoretic models in which electromagnetic interactions are mediated via operators that sample fields at multiple spacetime points, as governed by entire functions or spacetime correlation functions. This generalized interaction structure can be implemented at the fundamental level—by modifying the fermion and photon kinetic and vertex terms, the Feynman rules, and even the quantization prescription itself. The resulting nonlocality, while preserving local gauge invariance via path-ordered gauge links or entire function smearing, leads to profound consequences for ultraviolet behavior, renormalization, form factors, and phenomenology. Nonlocal QED can serve as an ultraviolet complete theory, a framework for modeling composite charges, a tool to explain anomalous lepton magnetic moments without new fundamental degrees of freedom, and a route to constructing effective interactions relevant for atomic spectroscopy, strong-field processes, and condensed matter settings.

1. Gauge Invariance and Nonlocal Interaction Structure

Gauge invariance at all orders remains the cornerstone for field-theoretic consistency in nonlocal QED. Instead of modifying the covariant derivative directly, nonlocal QED promotes the minimal coupling or vertex term

$e\bar\psi(x)\gamma^\mu \psi(x) A_\mu(x)$

to a form involving an entire function smearing operator: $$e\to eE(x), \quad \mathrm{so\ that\ } e\bar\psi(x)\gamma^\mu\psi(x)A_\mu(x) \to \bar\Psi(x)\gamma^\mu\Psi(x)A_\mu(x),\quad \Psi(x)=E(x)\psi(x)$$ (Moffat, 2011). The function $E(x)$ is chosen to be analytic and normalized on-shell ($E(0)=1$) to ensure that the interaction is local at large distances, while strongly suppressing high-momentum contributions.

To restore local $U(1)$ invariance, the explicit evaluation of the gauge variation shows that the smearing-induced “spurious” terms are cancelled by higher-order compensation terms in the nonlocal transformation law, yielding exact invariance to all orders (Moffat, 2011).

The most general construction replaces both the free and interaction terms: $\mathcal{L}_\text{nonlocal} = \int d^4a\, \bar\psi \left(x+\frac{a}{2}\right) F_1(a)\left(i\slashed{\partial} - m\right) \psi\left(x-\frac{a}{2}\right) - e \int d^4a\, d^4b\, \bar\psi\left(x+\frac{a}{2}\right) \slashed{A}(x+b) \psi\left(x-\frac{a}{2}\right) F_1(a) F_2(a,b) + \dots$ with path-ordered gauge links inserted between fields to guarantee local gauge invariance (Wang et al., 30 Sep 2025, Li et al., 16 Jan 2024). These links generalize the Wilson line, restoring invariance when fields at different points transform under local $U(1)$.

2. Ultraviolet Completion, Renormalization, and Finiteness

The primary motivation for nonlocal QED is the ultraviolet (UV) regularization delivered by the entire function form factors. The damping behavior of $E(p/\Lambda)$ yields UV-convergent loop integrals for all Feynman diagrams. For instance, the fermion self-energy and photon vacuum polarization become finite, irrespective of loop order, provided $E(p)$ decays sufficiently rapidly as $|p|\to\infty$: $$\Sigma(p) = -ie \int d^4k\, E(k-p) \gamma^\mu S(k) \gamma^\nu D_{\mu\nu}(p-k) E(k)$$

This finiteness extends to the renormalization constants,

$Z_1,\, Z_2,\, Z_3 < \infty$

and the running of the electromagnetic coupling $\alpha(Q)$ remains well-defined up to arbitrarily high scales, with the form factor $E(Q/\Lambda_\mathrm{EM})$ controlling the asymptotics (Moffat, 2011, Capolupo et al., 2023). The Källén–Lehmann representation of the photon propagator

$$D(k) = \frac{Z_3}{k^2} + \int_0^\infty dM^2 \frac{\sigma(M)}{k^2 - M^2 + i\varepsilon}$$

with $Z_3 + \int \sigma(M) dM^2 = 1$, develops a spectral density that reflects nonlocality. The result is a non-singular Coulomb potential at $r=0$: $$V(r) = e^2 \int \frac{d^3k}{(2\pi)^3} e^{i\vec{k}\cdot\vec{r}} D(k)$$ which is finite at the origin, eliminating the $1/r$ divergence of local QED (Moffat, 2011).

3. Modified Feynman Rules and Ward-Takahashi Consistency

Implementation of nonlocal QED at the perturbative level necessitates a systematic replacement in the Feynman rules:

• Propagators: $S(p) = i/(\slashed{p} - m) \cdot 1/E_1(p)$ for fermions, $D_{\mu\nu}(k) = -i g_{\mu\nu}/k^2 \cdot 1/E_4(k)$ for photons.
• Vertices: The standard $-ie\gamma^\mu$ becomes $-ie\gamma^\mu G_2(p,q)$, a function of the external and internal momenta and the nonlocal correlation functions.
• Gauge-link–induced vertices: Additional vertices arise from the path-ordered gauge link expansion, introducing extra diagrams even at one-loop order for processes such as the lepton $g-2$ (Li et al., 16 Jan 2024, Wang et al., 30 Sep 2025).

For example, the full one-photon vertex receives corrections: $V_1^\mu(p,q) = \gamma^\mu G_2(p,q),$

$V_2^\mu(p,q) = (\slashed{p}-m) \frac{(q+2P)^\mu}{(q+P)^2-P^2}\left[G_3(q+p,q)-G_3(p,q)\right]$

and higher-order photon emission vertices ($V_3^{\mu\nu}$, $V_4^{\mu\nu}$, etc.) are derived accordingly.

Despite these modifications, the Ward–Green–Takahashi identities

$$\lim_{q\to 0}[-i q_\mu \Gamma^\mu(p+q,p)] = S^{-1}(p+q) - S^{-1}(p)$$

are preserved order by order, provided the correlator normalization constraints are satisfied ($F_1(0)=1$) (Li et al., 16 Jan 2024, Wang et al., 30 Sep 2025).

4. Physical Observables and Phenomenology

a. Anomalous Magnetic Moments ($g-2$)

Nonlocal QED predicts corrections to the lepton magnetic moments arising from both modified propagators and the additional vertex diagrams from the gauge links. For the lepton Pauli form factor $F_2(0)$,

$a_l = \frac{g-2}{2} = F_2(0)$

the net contribution can explain the experimentally observed muon and electron $g-2$ anomalies for cutoff parameters ($\Lambda_1$, $\Lambda_2$) in the TeV range, without invoking any new particles (Li et al., 16 Jan 2024, Wang et al., 30 Sep 2025, He et al., 2019). For example, with $\tilde F_1(p) = (\Lambda_1^2 - p^2)/\Lambda_1^2$, and $$\tilde F_2(p) = \tilde F_3(p) = \Lambda_2^2/(\Lambda_2^2-p^2)$$, the additional gauge-link–induced diagrams and the modified vertex structure shift $a_\mu$ and $a_e$ by amounts that match experimental discrepancies.

b. Lamb Shift and Precision Tests

Nonlocal modifications to the photon propagator introduce new contact terms in the Coulomb potential: $$V(r) = -\frac{e^2}{4\pi r} - \frac{e^2}{M_\gamma^2}\delta^3(r) + \cdots$$ yielding shifts in hydrogenic energy levels, particularly magnified in muonic hydrogen (by a factor $\sim (m_\mu/m_e)^3$), potentially accommodating the 0.311 meV discrepancy in the Lamb shift (muonic hydrogen anomaly) and enabling phenomenological bounds on the nonlocality scales (Capolupo et al., 2023).

c. High-Energy Cross Sections and Running Coupling

Proton–proton Drell–Yan production cross sections exhibit deviations from local QED at invariant masses $Q\gtrsim \Lambda_{EM}$ ($\sim 1-2\;\mathrm{TeV}$) (Moffat, 2011). The running of the QED coupling $\alpha(Q)$ acquires extra $Q^2$-dependent and nonlocal logarithmic corrections, subordinate at low energies but potentially significant at high momentum transfers (Capolupo et al., 2023).

5. Quantization and “Solid Quantization” Paradigm

Nonlocal theories necessitate a modification of the canonical quantization procedure: the usual equal-time commutator

$$[\phi(\vec{x},t), \pi(\vec{y},t)] = i \delta(\vec{x}-\vec{y})$$

is replaced with

$$[\phi(\vec{x},t), \pi(\vec{y},t)] = i \Phi(\vec{x}-\vec{y})$$

where $\Phi(x)$ is a smearing function normalized to unity, consistent with the underlying nonlocal correlation functions (Wang et al., 30 Sep 2025). In momentum space, this induces a replacement

$$[a_{\vec{p}}, a_{\vec{q}}^\dagger] = \Psi(\vec{p}) \delta^{(3)}(\vec{p} - \vec{q})$$

with $\Psi(\vec{p})$ the Fourier transform of $\Phi$, entering directly into the propagators. This “solid quantization” approach yields the same modifications as the nonlocal Lagrangian approach and preserves causality and unitarity.

6. Comparison with Local and Other Nonlocal QED Extensions

Nonlocal QED provides a self-consistent UV completion by replacing the need for the Higgs mechanism in the QED sector, producing finite loop amplitudes, and enabling a spectral Källén–Lehmann representation with smeared short-distance behavior (Moffat, 2011). In macroscopic QED, nonlocal response functions and the relaxation of Onsager reciprocity have been treated within a Green tensor framework, ensuring duality invariance even in nonreciprocal media (Buhmann et al., 2011). In condensed matter and cold atom applications, nonlocal-proca and pseudo-QED models have been constructed to capture mixed-dimensional effects and short-range Yukawa screening (Alves et al., 2017, Alves et al., 2019).

7. Applications and Outlook

Nonlocal QED frameworks have found application in:

• Atomic and molecular physics: explaining g-2 anomalies, and modifying Lamb shifts via nonlocal corrections to the Coulomb potential (Capolupo et al., 2023, Wang et al., 30 Sep 2025).
• Metrological and precision spectroscopic contexts: providing more accurate QED corrections in heavy ions through nonlocal potentials (Tupitsyn et al., 2016).
• Collider phenomenology: predicting new scaling behaviors in high-energy cross sections that are accessible at the LHC (Moffat, 2011).
• Quantum information: engineering entanglement purification protocols for nonlocal microwave photons in circuit QED, with immediate utility for quantum communication (Zhang et al., 2017).
• Nonlocal field theory model-building: constructing UV-complete or conformally invariant nonlocal QED models with vanishing beta function (Heydeman et al., 2020), and realizing new perspectives on the quantum–information–theoretic structure of gauge theories in the algebraic AQFT framework (Twagirayezu, 19 Jul 2025).

Future directions involve systematic exploration of solid quantization in curved spacetimes, detailed phenomenological fits to lepton g-2 anomalies without new particles, nonlocal extensions to non-Abelian gauge theories, and exploitation of nonlocal regularization as a foundation for strongly coupled effective field theories.

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Table: Key Nonlocal QED Features and Their Physical Implications

Feature	Formulation/Equation	Physical Consequence
Entire function/Correlation function $E$	$e\to eE(x)$, $\psi\to E(x)\psi$	UV finiteness; vertex damping
Gauge link operator	$I(x,y) = \mathcal{P}\exp\{\dots\}$	Local $U(1)$ invariance
Modified propagator	$S(p) = i/(\slashed{p}-m) E(p)^{-1}$	Regularized loop diagrams
Extra gauge-link–induced vertices	e.g., $V_2^\mu(p, q)$, $V_3^{\mu\nu}$	Additional diagrams in g-2, other amplitudes
Källén–Lehmann spectral density	$D(k) = Z_3/k^2 + \cdots$	Smeared short-distance Coulomb; finite $r=0$
“Solid quantization”	$[\phi(x),\pi(y)] = i \Phi(x-y)$	Modified field commutator structure

This synthesis demonstrates that nonlocal QED, properly constructed, yields UV-complete, gauge-invariant, and phenomenologically flexible extensions of conventional quantum electrodynamics, with distinctive predictions for high-energy, atomic, and quantum information settings.