UV-finite Effective Field Theory from Quantized Irreversible Null-geometry

Abstract: Standard perturbative quantum field theory is persistently challenged by ultraviolet (UV) divergences and overlapping sub-divergences. In this work, we construct an absolutely UV-finite effective field theory (EFT) natively in four-dimensional spacetime. We embed the complete dynamical operator within a double-exponential capacity measure derived from quantized irreversible null-geometry. By analytically expanding the exact operator resolvent, we demonstrate that a universal macroscopic proper-time bound ($T \ge τ0$) natively emerges, structurally collapsing the high-order UV phase space via topological contraction. This mathematically defines the absolute UV boundary of the EFT. Consequently, overlapping subgraph divergences are rigorously identified as distributional pseudo-singularities caused by the continuous approximation analytically oversmoothing the discrete lattice bounds. These artifacts are systematically corrected by employing the Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) protocol strictly as an algebraic mapping operator. Evaluating Quantum Electrodynamics in 4D, we obtain a finite electron bare mass ($m_0 \approx 0.427$ MeV) and derive finite renormalization constants, analytically preserving the exact Ward-Takahashi identity ($Z_1=Z_2$). In Quantum Chromodynamics, the exact preservation of Slavnov-Taylor identities yields a calculable finite bare strong coupling ($α{s,0} \approx 0.0191$). Generating analytically closed non-perturbative Schwinger-Dyson Equations, this framework provides a consistent, anomaly-free mathematical foundation for finite effective field theories to address dynamic chiral symmetry breaking and mass generation.

• The paper introduces a novel effective field theory that replaces conventional continuous path integrals with a double-exponential measure from a discrete spacetime, ensuring absolute UV convergence.
• It employs an operator formalism that redefines gauge interactions in QED and QCD, yielding finite bare masses and charges while preserving key symmetries without counterterms.
• The framework extends to non-perturbative dynamics, enabling analytic treatment of chiral symmetry breaking and color confinement through a bounded proper-time evolution.

UV-Finite Effective Field Theory from Quantized Irreversible Null-Geometry

Framework Motivation and Mathematical Architecture

The persistent challenge of ultraviolet (UV) divergences in standard four-dimensional quantum field theory (QFT) is addressed by constructing an effective field theory (EFT) grounded in a discrete spacetime substrate described by quantized irreversible null-geometry (QIN). By replacing the conventional continuous path integral measure with a double-exponential statistical capacity, derived from a Poisson point process at the Planck scale, the framework introduces a fundamental scale—characterized by the proper-time interval $\tau_0$—imposed natively via operator calculus. The mapping leads to a non-linear dynamical operator whose resolvent is analytically expanded, resulting in a discrete, summable representation of proper-time evolution. This enforces absolute convergence in the UV, rigorously bounding loop integrals by a universal macroscopic limit $T \geq \tau_0$.

Geometric discreteness restricts the physical vacuum to a sparse, stochastic point process, and the double-exponential measure structurally recovers low-energy physics while strictly delimiting high-frequency fluctuations. The framework achieves a strict separation between global physical divergences, intercepted by the topological contraction mechanism, and distributional subgraph artifacts, which are proven to arise solely from analytically over-smoothing the discrete geometry via the continuum approximation.

Operator Formalism and Topological Contraction

The EFT abandons elementary kinetic-interaction partitioning, formulating all regularization within a Schwinger proper-time operator formalism. The dressed inverse Green's operator is defined via a double-exponential in capacity measure, ensuring the recovery of standard QFT at sub-Planckian scales, while preventing access to trans-Planckian artifacts. Analytic expansion yields a discrete series; each term enforces Hilbert-Schmidt suppression of UV modes, collapsing multi-loop internal structures into singular contact interactions. Topological contraction at the UV limit is shown to be an inevitable algebraic consequence, not a heuristic regularization.

Subgraph divergences, commonly appearing in continuum QFT as fractional $\epsilon$-poles, are rigorously re-interpreted as pseudo-singularities. They are artifacts introduced when mapping discrete sums to the Riemann integral—enabling unphysical overlap of interaction vertices. The BPHZ protocol is recast as a purely algebraic subtraction operator acting on the continuous simplex, excising distributional artifacts while strictly preserving physical gauge symmetries.

Abelian and Non-Abelian Gauge Theory Results

The framework is explicitly applied to Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD), evaluating standard loop corrections in Minkowski spacetime. All primitively divergent QED amplitudes are shown to converge natively in $d=4$:

• Finite electron bare mass and renormalization: The geometric evaluation yields a bare electron mass $m_0 \approx 0.427\,\text{MeV}$ and preserves the Ward-Takahashi identity exactly ($Z_1 = Z_2 \approx 0.94$), eliminating the need for negative divergent mass counterterms.
• Charge renormalization: The photon vacuum polarization yields a finite $Z_3 \approx 0.92$. The bare coupling is anchored as a finite geometric constant, $e_0 = 1.04 e_{\text{obs}}$.
• Two-loop QED: The framework supports analytic evaluation of overlapping divergences, with two-loop corrections matching expected $\beta$-function coefficients and yielding convergence solely via the universal global boundary.

For non-Abelian QCD:

• Exact preservation of BRST and Slavnov-Taylor identities: The operator measure is proven to commute with all gauge symmetries. The framework maintains nilpotency and algebraic cancellation, natively in $d=4$, without counterterms.
• Finite absolute non-Abelian bare charge: The running coupling is reformulated without the sliding scale $T \geq \tau_0$0, yielding an absolute screening relation. Anchoring at the $T \geq \tau_0$1 boson mass scale and Planck boundary, the bare strong coupling evaluates to $T \geq \tau_0$2—well within the perturbative regime.
• Dimensional consistency and gauge covariance: All loop integrals are executed in unconstrained momentum space, with proper-time boundary acting as a scalar cutoff. Anomalous surface terms are entirely absent; asymptotic freedom is recovered with group-theoretic coefficients as in standard QCD.

Non-Perturbative Dynamics and Chiral Symmetry Breaking

The analytic extension to non-perturbative Schwinger-Dyson Equations (SDEs) proceeds naturally, regularizing the kernel with global proper-time bound $T \geq \tau_0$3. The discrete measure enables evaluation of exact gap equations and non-perturbative propagators as Fredholm integral equations. Crucially, the operator measure does not exponentiate dimensionful Dirac operators, avoiding spectral pathologies endemic to heuristic cutoffs. Dynamic chiral symmetry breaking (DCSB) and fermion mass generation emerge from the macroscopic topological boundary, enforcing finite mass gap solutions without triviality or anomalous symmetry breaking.

For the Yang-Mills sector, the framework prevents the introduction of bare gluon mass and algebraically locks the polarization tensor into the transverse form, enabling color confinement via the Schwinger mechanism. The dynamical gluon mass runs naturally towards the non-trivial infrared pole, with macroscopic scale set by geometric capacity.

Implications and Future Directions

Formulating QFT regularization strictly in operator calculus within a topological geometric substrate provides strong mathematical guarantees for UV finiteness, absolute convergence, and exact symmetry preservation in $T \geq \tau_0$4. The framework obviates the need for dimensional analytic continuation, arbitrary sliding scales, and symmetry-restoring counterterms, establishing a highly consistent foundation for both perturbative and non-perturbative quantum dynamics.

Practically, this suggests direct applicability in precision calculations at both low and high energy, enabling systematic evaluation of multi-loop corrections without renormalization ambiguities. Theoretically, it opens avenues for rigorously investigating quantum gravity, cosmological vacuum structure, and strong interaction phenomena, including mass gap generation and color confinement, via strictly bounded operator measures.

Future developments may explore the incorporation of interacting gravity sectors, the role of geometric stochasticity in quantum cosmology, and applications in lattice field theory where the discrete proper-time bound may inform algorithmic efficiency and sampling. Adaptation to more general topological backgrounds or higher-dimensional extensions could reveal additional regularization architectures.

Conclusion

The quantized irreversible null-geometry framework establishes an absolutely UV-finite, divergence-free effective field theory in four-dimensional Minkowski spacetime. The operator formalism, grounded in discrete topology, enforces universal proper-time bounds, guarantees absolute convergence, and preserves exact gauge symmetries across Abelian and non-Abelian sectors. Analytical extension into the non-perturbative regime provides robust mechanisms for chiral symmetry breaking and mass gap generation, without the pathologies of conventional regularization. This mathematically rigorous approach offers a unified and consistent structure for quantum field theory, with significant implications for both fundamental theory and applied computation (2606.22788).