Ultraboosted Gauge Potentials

• Ultraboosted gauge potentials are gauge fields examined under extreme Lorentz boosts, exhibiting shockwave behavior and distributional singularities.
• They reveal distinct properties across classical electromagnetism, quantum field theory, and M-theory, elucidating nonlocal interactions and gauge invariance.
• Advanced methods like dimensional regularization and nonlocal form factors ensure UV and IR finiteness, underpinning robust high-energy theoretical models.

The ultraboosted limits of gauge potentials concern the mathematical and physical characterization of gauge fields—particularly electromagnetic and nonabelian potentials—under extreme Lorentz boosts, or in high-energy, high-momentum transfer regimes. The investigation into ultraboosted gauge potentials addresses their transformation properties, singularities, distributional behavior, and constraints arising from gauge invariance and symmetry principles. A collection of recent works has clarified this landscape in classical electromagnetism, quantum field theory, quantum condensed matter, and M-theory.

1. Classical Electromagnetism: Shockwave Structure and Null Defects

The behavior of electromagnetic gauge potentials in the limit of infinite boost has been long-standingly problematic. The canonical Liéard–Wiechert solution yields, for a particle with finite boost velocity $u$, a potential

$A_u^\mu = -\frac{g F^2}{4\pi} \frac{v^\mu}{R_u}$

where $R_u^2 = (u x^0 + x^1)^2 + (1-u^2)x_\perp^2$. In the ultraboost limit ($u\to 1$), field strength localizes into a shockwave: $$F_{+i} = \frac{g F^2}{2\pi}\,\frac{x_i}{|x_\perp|^2} \,\delta(x^+)$$ with $x^+$ the lightlike coordinate. The gauge potential becomes distributional,

$$A^- \sim \frac{gF^2}{2\pi} \left[ \log(\mu^2 x_\perp^2)\,\delta(x^+) - \frac{1}{|x^+|} \right]$$

where the second term is pure gauge. Only the shockwave field strength is physically meaningful (Erramilli et al., 4 Sep 2025).

Null line defects (null Wilson lines) serve as the correct proxy for ultraboosted charged particles. Their extended nature and inherent Lorentzian signature allow for a rigorous distributional interpretation, supported on the restricted subspace $\mathcal{S}_{\rm null}(\mathbb{R}^4)$ of Schwartz test functions. Maximal conformal symmetry, preserved by null defects, further constrains correlators—resulting in vanishing one-point functions up to delta function singularities—and ensures trivialization except for purely contact-type terms. The perfect null polygon emerges as the unique ultraboosted charge configuration consistent with Gauss’ law on compact spatial manifolds (Erramilli et al., 4 Sep 2025).

2. Gauge Transformations, Invariance, and Velocity Gauges

Gauge invariance in electrodynamics manifests in two principal concepts: the symmetry of the Lagrangian (photon as gauge particle) and the non-uniqueness of potentials representing E and B fields (Reiss, 2021). Potentials are more fundamental than fields—all equations of motion, quantum and classical, are written in terms of potentials, not directly in terms of E or B, and the Aharonov–Bohm effect demonstrates phase sensitivity even in regions with vanishing field.

Under ultraboost, only gauge transformations preserving the underlying symmetries and the lightcone (i.e. $A^\mu \propto f(k\cdot x)$ with $k^\mu$ null and $\Lambda$ a function of $k\cdot x$) are admissible. Any transformation violating this constraint disrupts physical conservation laws and propagation properties. The velocity gauge (Giri et al., 2022) generalizes both Lorenz ($v=c$) and Coulomb ($v\to\infty$) gauges; ultraboosted limits in gauge speed ($v\gg c$) lead to non-causal potentials, but physical fields remain causal and invariant after exact cancellation of acausal components.

Gauge Class	Propagation Speed	Field Invariance
Lorenz	$v = c$	Causal, invariant
Coulomb	$v \rightarrow \infty$	Instantaneous, invariant
Velocity (generic)	$v \in \mathbb{C}-\{0\}$	Fields always invariant

3. Point-Localized vs String-Localized Potentials

Attempts to regularize gauge theories using string-localized potentials constructed from gauge invariant observables have improved UV properties but fail to fully encode essential nonlocal features. String-localized potentials, defined as

$$A^{\text{string}}_\mu(x,e) \sim \int_0^1 du\, F_{\mu\nu}(x + ue)\,e^\nu$$

do not produce nontrivial automorphisms necessary for gauge bridges—structures that enforce Gauss’ law and encode long-range interactions between charges. Specifically, automorphisms generated solely from observable-based limits act trivially on flux operators, and thus cannot fully describe sectors with net charge (Buchholz et al., 2019). Poincaré invariant theories constructed from such potentials are not equivalent to conventional QED, as they lack longitudinal photon degrees of freedom needed to ensure correct flux quantization.

4. Quantum Gauge Theories: Regularized Ultraboosted Limits

In higher-dimensional models, as in gauge-Higgs unification on $M^4\times S^1$, the zero mode of the extra component of the gauge potential serves as the 4D Higgs field (0709.2844). Two-loop calculations, using dimensional regularization, yield finite Higgs mass and potential: $$m_H^2 = \frac{9e_4^2 \zeta_R(3)}{16\pi^4 R^2} \left[1 - \frac{e_4^2 \ln 2}{24\pi^2}\right]$$ Ultraboosted configurations along the extra dimension do not generate divergences—protective mechanisms inherent to gauge-Higgs unification control both UV and scheme dependence, making these frameworks robust under high-momentum probes and ultraboosted gauge configurations.

Nonlocal quantum gauge theories (Modesto et al., 2015) utilize entire function form factors in the kinetic term: $$\mathcal{L}_{\rm gauge} = -\frac{1}{4g^2} \operatorname{tr}\Big[F\,e^{H(\Box/\Lambda)}F + V_{\rm pot}(F)\Big]$$ Dressed propagators exhibit exponential suppression in the UV,

$$O^{-1}(k) \sim \frac{e^{-H(k/\Lambda)}}{k^{2\gamma+D-2}}$$

and both Landau pole and IR singularities are eliminated. Even under ultraboosted (high-energy) limits, gauge potentials remain regular, and the UV/IR finiteness is preserved, providing a consistent framework for the perturbative treatment of non-Abelian gauge theories across all energy scales.

5. Ultraboosted Gauge Potentials in Condensed Matter, Gravity, and M-Theory

In synthetic gauge systems, ultraboosted limits are replicated via strong gauge potentials, rapid lattice tilting, or high tunneling rates, restructuring band spectra and enabling access to topologically nontrivial phases (Mandal et al., 2016). In hydrodynamic Bose condensates with density-dependent gauge potentials (Buggy et al., 2017), the explicit breakdown of Galilean invariance underlines the novel nonlinear effects induced by ultraboosted flows; kinetic energy acquires flow- and density-dependent corrections, altering pressure and excitation spectra.

Gravitational models formulated via non-Abelian gauge potentials (e.g., $U(2)$-based torsionless connections) translate extreme boosts in the gauge sector into shockwave gravitational metrics, potentially connecting with Aichelburg–Sexl-type solutions (Minguzzi, 2014).

In the context of M5 branes and 11-dimensional supergravity, global completion of gauge potential data is achieved via non-abelian cohomotopy (flux quantization), wherein traditional local formulas arise as “ultraboosted limits” of global null-concordance interpolations (Banerjee, 9 Jul 2025). This formulation ensures that gauge potentials and transformations conform to modified Bianchi identities and patching conditions on curved and orbifold spacetimes, even as fluxes are ultraboosted from trivial to physical states via the homotopy parameter.

6. Integrable and Nonintegrable Dynamics: Solitons and Ultraboost

Spinor BECs with matrix gauge potentials display a Zeeman-controlled crossover between integrable and nonintegrable dynamics (Kartashov et al., 2019). Absent Zeeman splitting, the system is gauge equivalent to the Manakov model; ultraboosted (large $\Omega$) limits restore integrability by averaging out noncommutative perturbations. The explicit singular (ultraboosted) regime is thus regularized at the level of the effective field theory.

7. High-Energy Phenomena and Collider Physics

Ultraboosted top quark production at collider energies enables stringent probes of flavor-changing gauge couplings. Energy-enhanced dipole operators dominate in the ultraboosted kinematic regime, with branching ratios for $t\to uZ$ and $t\to u\gamma$ constrained to $10^{-5}$ at HL-LHC and $10^{-6}$ at FCC-hh (Aguilar-Saavedra, 2017). Effective field theory techniques are essential for parameterizing gauge potential contributions and interpreting rate limits in terms of physics beyond the Standard Model.

Summary Table: Ultraboosted Limit Features Across Contexts

Domain	Characteristic Ultraboosted Behavior	Mathematical/Physical Resolution
Classical EM	Shockwave, delta function fields	Distributional interpretation, null defects (Erramilli et al., 4 Sep 2025)
Quantum Theory	UV/IR finiteness, regular propagators	Nonlocal form factors; protected by unification (Modesto et al., 2015, 0709.2844)
Gauge Bridges	Missing in string-localized observable potentials	Only fully local fields capture Gauss’ law (Buchholz et al., 2019)
Gravity	Shockwave metrics, background expansion	Gauge/gravity correspondence, metric from gauge potential (Minguzzi, 2014)
M-Theory	Global ultra-extensions, null-concordance	Cohomotopy, flux interpolation (Banerjee, 9 Jul 2025)

Conclusion

The paper of ultraboosted limits of gauge potentials reveals deep interconnections between distributional mathematics, conformal symmetry, gauge invariance, physical regularization, and topological quantization. Across domains—electromagnetism, gauge theory, gravitation, string theory, and condensed matter—the correct understanding of ultraboosted gauge fields requires careful attention to distributional definitions, admissible gauge transformations, global completion of transition data, and the physical significance of symmetry constraints. These insights clarify longstanding questions (such as shockwave solutions in electrodynamics and the nonlocality of observable-based potentials), and enable robust formulations of quantum and geometric field theories resilient to singular boosts and high-energy deformations.