I. UV Divergence Is Not a Natural Property but a Modeling Error
In the context of Natural Quantum Theory (NQT), "UV divergence" in quantum theory is not an intrinsic property of the physical world, but a product of modeling and formulation:When we assume a system is of infinite domain, zero size, zero dissipation, driven by white noise, and pursue "absolute quantities" (rather than observable differences), the high-frequency end of spectral integrals will inevitably go wrong.
In other words:UV divergence does not reveal "infinite energy in nature," but reflects the mathematical pathology of "models lacking physical constraints."
This problem is particularly prominent in Quantum Field Theory (QFT):Traditional QFT treats fields as point interaction systems defined in unbounded flat spacetime, leading to infinite mode density and unconstrained spectral integrals. As a result, "divergence" is no longer a physical phenomenon, but a product of "unbounded spectral representation."
II. Why Other Spectral Disciplines Are "Naturally Free of Divergences"
In disciplines with mature spectral analysis—such as Fluid Mechanics, Plasma Physics, Acoustics, Electromagnetism, and Elasticity—divergences almost never occur.The reason is that these fields incorporate real physical constraints from the very start of modeling:
| Physical Constraints | Examples | Impact on Spectral Integrals |
|---|---|---|
| Dissipation | Fluid viscosity, Landau damping, collision frequency | Exponential decay of high-frequency spectra, integral convergence |
| Boundaries | Pipe walls, cavities, conductors, magnetic confinement regions | Modal discretization or transformation into integrable forms |
| Finite energy and regularity | Initial/boundary conditions belonging to L²/H^s spaces | Elimination of high-k singular modes |
| Finite correlation length | Molecular mean free path, Debye length | Natural weakening of high-k responses |
| Finite-bandwidth driving | Real external fields are non-white noise | Rapid attenuation of high-frequency weights |
| Calculation of only observable differences | Spectral flux, drag change, modal energy exchange rate | Differences are naturally finite and measurable |
Thus, "UV divergence" is not an inherent risk of spectral methods, but an illusion arising when spectral methods are detached from physical constraints.
III. Where Does "UV Divergence" Come From: Five Root Causes of the Model
1. Zero size/point interactions
Treating particles as geometric points makes coupling vertices equivalent to δ-source convolution. All high-k weights are constant, so the integral ∫kⁿdk must diverge.
2. Unbounded/infinite domain
Mode density ~k², modal energy ~ω(k), and their product grows with Λ⁴, with no geometric cutoff conditions.
3. Pursuit of "absolute quantities" rather than "differences"
Calculating the absolute value of zero-point energy in the entire space is certainly divergent; the Casimir difference, however, is finite and measurable.
4. Idealization of white noise/instantaneous propagation
The δ-correlation assumption flattens high-frequency weights; real systems always have finite bandwidth and time delay.
5. Neglect of finite correlation scale
After introducing the form factor F(k)∼e^{-(ka)²}, the integral converges automatically—this is the frequency-domain expression of finite structures.
IV. Three "Hemostatic Principles" of Natural Quantum Theory
NQT systematically rejects the above root causes and proposes three fundamental principles to make spectral integrals naturally finite:
1. Constrained Spectrum Principle
The "eigenfrequency spectrum" can only be discussed in systems with geometric, medium, or symmetry constraints. The unconstrained vacuum is not assigned absolute energy; only spectral differences caused by boundary changes are calculated.
2. Finite Scale Principle
Elementary particles (electrons, neutrinos, etc.) have finite rotation-circulation structures, and current density inherently carries a form factor F(k), so high-k power divergences are physically cut off.
3. Observable Difference Principle
All measurable quantities are differences: energy level differences, Casimir force, scattering cross-section corrections. Treating constant terms as "intrinsic energy" is a category error.
The physical significance of these three principles is:"Divergence" is not an abyss of nature, but a modeling error. Correcting the model rather than renormalizing the results is the scientific direction.
V. One-to-One Correspondence with Fluid and Plasma Physics
| Common QFT Operations | Physical Counterparts in Fluid/Plasma Physics | Physical Meaning |
|---|---|---|
| "Dressing," "polarization," "running coupling" | Viscous dissipation, Landau damping, Debye screening | High-frequency modes are hindered, spectra converge |
| "Unbounded space integration" | Actual boundary conditions | Discrete spectra in finite domains |
| "Point particles" | Finite vortex cores or gyroradii | Geometric form factors suppress high-k |
| "Vacuum zero-point energy" | Energy differences, flux differences | Only differences are measurable |
| "White noise external sources" | Finite-correlation driving | Actual physical bandwidth is finite |
It is evident that NQT’s handling of spectral integrals is fully consistent with the principles of spectral analysis in engineering physics:Once real physical boundaries and finite scales are retained, the divergence problem naturally disappears.
VI. Two Types of "Divergence Diseases" and Physical Prescriptions
(1) Symptoms and Prescriptions for Zero-Point Energy Divergence
Symptoms:ρ∼∫d³k (1/2)ℏωₖ∝∫^Λk³dk→Λ⁴
Prescriptions:Introduce the form factor F(ka) (a is the correlation length).If F(x)∼e^{−x²}, then ρ∼a⁻⁴ is finite;Only discuss the boundary-induced change Δρ, which is measurable and geometrically reversible.
(2) Symptoms and Prescriptions for Scattering/Vertex Divergences
Symptoms: Loop integrals of point coupling vertices diverge.Prescriptions:Convolutionalize the current density: J(x)→∫fₐ(x−y)J(y).Multiply by F(ka) in momentum space to automatically cut off high-k.This is not a trick, but a geometric reflection of the finite structure of particles.
VII. Unification of Mathematical Structure and Physical Intuition
From the perspective of spectral theory, the root cause of divergence lies in unbounded spectral integrals:
Eᵥₐc = (1/2)∫₀^∞ℏω(k)D(k)dk, D(k)∝k²
If ω(k)=ck, then ∫k³dk diverges. However, "infinite k" modes have never existed in reality.Any real system—fluids, plasmas, solids, electronic fields—has a natural cutoff.Thus, the meaning of renormalization is to artificially supplement this natural cutoff.
VIII. Operational Checklist: How to Physicize Spectral Modeling
IX. Conclusions and NQT’s Attitude
"UV divergence" is not an abyss of nature, but a modeling error.When we mistake unboundedness for boundaries, point sources for structures, and absolute quantities for observables, divergences arise.Returning to "constrained spectrum—finite scale—measurable differences" makes them naturally disappear; what remains are physical problems, not mathematical phantoms.
Decades of spectral method practice in fluid and plasma physics have proven:With real boundaries, dissipation, and finite correlation lengths, spectral integrals are tame and powerful.NQT merely systematically brings this common sense back to the quantum world.