Natural Elimination of Ultraviolet Divergence: Based on Frequency-Difference Transition Amplitudes and Discrete Mode Spectra

Author: Zhang Suhang, Luoyang, Henan

Abstract

Ultraviolet divergence in quantum field theory is a long-standing problem that theoretical research needs to address. Existing regularization and renormalization methods are mostly external mathematical treatments, not physical laws derived from the theory itself. In scenarios involving mode coupling transitions, this paper presents two independent paths of natural elimination: first, the transition probability between modes decays as the frequency difference increases; when the decay rate satisfies certain conditions, the loop integral naturally converges. Second, the mode spectrum itself is discrete; when the number of modes is finite or the frequency has an upper limit, the summation naturally converges. The two paths have different starting points and different conditions, and can hold independently. If either one is satisfied, the ultraviolet divergence is eliminated. This paper also distinguishes two types of physical models, clarifying that the completely uncoupled free field is merely an idealized construct, and the zero-point fluctuation divergence under that model is not covered by the paths in this paper.

Keywords: ultraviolet divergence; frequency difference; transition amplitude; discrete mode spectrum; natural elimination; idealized model

I Introduction

Quantum field theory has achieved great success in describing microscopic particle interactions, but the problem of ultraviolet divergence still needs to be addressed. Under the traditional framework, the field is decomposed into infinitely many normal oscillation modes; when summing or integrating the contributions of high-frequency modes, an infinity arises, and finite observable quantities cannot be directly obtained.

To handle the divergence, researchers have developed methods such as momentum cutoff, dimensional regularization, Gaussian regularization, and lattice field theory. These methods share a common feature: they all introduce additional mathematical settings during the calculation. Momentum cutoff directly limits the highest momentum and breaks symmetry; dimensional regularization temporarily switches to a non-integer dimensional space and is merely an intermediate computational device; Gaussian regularization adds a decay factor; lattice field theory discretizes spacetime and is mostly used for numerical computation. These methods can serve as computational tools, but they are not deductions from fundamental theory.

In scenarios involving mode coupling transitions, this paper presents two independent paths of natural elimination. Neither path relies on artificial cutoffs, nor does either modify the dimensionality of spacetime.

II Traditional Regularization Schemes and Their Characteristics

2.1 Several Mainstream Regularization Methods

Existing methods for handling ultraviolet divergence are all external mathematical treatments:

1. Momentum cutoff: sets an upper limit on momentum and discards modes above that scale. The advantage is intuitive; the disadvantage is that it breaks Lorentz invariance and gauge symmetry.
2. Dimensional regularization: temporarily changes four-dimensional spacetime to d = 4 − ε dimensions, completes the integral in non-integer dimensional space, separates out the divergent pole, and finally returns to four dimensions. This method preserves symmetry and is a commonly used tool in analytical calculations, but it is merely a temporary construct used for computation.
3. Gaussian regularization: adds a Gaussian-type decay factor to suppress high-frequency contributions, which is an external weight.
4. Lattice field theory: discretizes spacetime into a grid, so the degrees of freedom are naturally finite, suitable for numerical simulation, but it breaks continuous spacetime symmetry.

None of the above methods are rules inherent to the physical system itself; they are computational aids introduced to obtain finite results.

2.2 Cognitive Problems Arising from Idealized Models

Traditional field theory often introduces the pure free-field model as a theoretical starting point. This model facilitates simplified analysis, but it is an abstract model stripped of all interactions. If this model, which is only applicable in a finite energy range, is extrapolated to the infinite high-frequency limit, divergence arises. It is necessary to distinguish mathematically constructed models from real physical processes in the world.

III Two Independent Paths of Natural Elimination

3.1 Basic Assumptions

In real physical systems, field modes often have coupling and transitions between them. This paper adopts the following relation: the transition amplitude between field modes i and j is determined by their frequency difference:

A_ij = F(Δν_ij)

Where A_ij is the transition amplitude, Δν_ij is the frequency difference between the two modes, and the transition probability P_ij = |A_ij|².

3.2 Path One: Amplitude Decay Makes the Integral Naturally Converge

The corresponding integral form in traditional field theory:

∫₀^∞ f(ω) dω

Without high-frequency suppression, the integral exhibits ultraviolet divergence.

For processes involving mode transition coupling, a probability weight corresponding to the frequency difference can be introduced, rewriting the integral as:

∫₀^∞ f(ω) · P(Δν) dω

The physical picture is: the larger the frequency difference between two modes, the lower the transition amplitude, and the corresponding contribution decays accordingly.

Convergence condition: when Δν → ∞, P(Δν) decays fast enough that the above generalized integral converges.

A specific example is P(Δν) ~ 1/(1 + (Δν)²). When Δν is large, P(Δν) ~ 1/Δν², which is sufficiently fast decay.

When this asymptotic decay condition is satisfied, the loop integral corresponding to the coupling transition yields a finite result, and the ultraviolet divergence is naturally eliminated.

This weight comes from the underlying relation of the transition amplitude, not from a patch added specifically to eliminate divergence.

3.3 Path Two: Discrete Mode Spectrum Makes the Summation Naturally Converge

Path one depends on the decay form of the amplitude with frequency difference. Path two does not depend on this; it starts from the structure of the mode spectrum itself.

Traditional field theory takes mode frequencies as continuous values, with infinitely many modes at the high-frequency end:

∫₀^∞ f(ω) dω

The divergence comes from the infinite number of modes.

If the mode spectrum is discrete:

ω₁, ω₂, ω₃, …

Then the summation form is:

Σ_n f(ω_n)

There are three cases for a discrete spectrum, any one of which makes the summation naturally converge:

Case One: Finite number of modes

The total number of modes is N, with N finite.

The summation is a finite sum:

Σ_{n=1}^{N} f(ω_n)

A finite sum yields a finite result, and the ultraviolet divergence is naturally eliminated.

Case Two: Infinite number of modes, frequency has an upper limit

The number of modes is infinite, but frequency can only take values up to some maximum ω_max.

The summation is:

Σ_{n=1}^{∞} f(ω_n), ω_n ≤ ω_max

The high-frequency end is bounded by the upper limit, and the summation naturally converges.

Case Three: Infinite number of modes, frequency spacing grows

The mode frequencies are:

ω_n ~ n^α, α > 0

The spacing between adjacent modes grows with n, and the mode density at the high-frequency end decreases.

As the mode density decreases, the summation naturally converges.

3.4 Sources of the Discrete Spectrum

The discrete spectrum is not an external assumption; it has three natural sources:

Source One: Finite volume

In a system of finite size, boundary conditions lead to discrete modes. Infinite space gives a continuous spectrum; finite space gives a discrete spectrum.

Source Two: Underlying discrete structure

If spacetime or the field itself is discrete, the modes are naturally discrete.

Source Three: Interactions lead to discrete energy levels

Systems with interactions generally have discrete energy levels. The free field has a continuous energy spectrum because it has no interactions.

3.5 Independence of the Two Paths

The two paths have different starting points and conditions:

Path One Path Two
Starting point Transition probability decays with frequency difference Mode spectrum is discrete
Key condition P(Δν) decays fast enough Number of modes finite or frequency has an upper limit
Mathematical form Integral weight suppresses high frequencies Summation itself is finite
Dependent object Amplitude form Spectrum structure

The two paths can hold independently.

· If the amplitude decays, path one holds;
· If the spectrum is discrete, path two holds.

If either one is satisfied, the ultraviolet divergence is naturally eliminated.

When both hold simultaneously, convergence is more stable.

IV Theoretical Applicable Boundaries and Discussion of Idealized Models

4.1 Distinction Between Two Types of Field Models

This paper divides field fluctuation models into two types:

1. Case One: Existence of mode coupling transitions

In real physical systems, weak coupling and transitions between different modes generally exist. Such scenarios are applicable to the two paths of natural elimination in this paper.

2. Case Two: Pure single-point free field

This model assumes that the field is completely isolated and that there are no transitions between modes. This model retains infinitely many independent high-frequency normal modes, and the zero-point energy summation still diverges.

4.2 The Difference Between Ideal Models and Real Physics

The completely interaction-free free field is a limiting model abstracted in theoretical research; there is no absolutely isolated field without any interaction in nature.

It should be noted that "idealized" does not mean "useless." The free-field model remains a fundamental tool in perturbation theory and the interaction picture. What this paper opposes is extrapolating this tool model to a limit where it does not hold, and attributing the resulting divergence to a property of nature.

Within the pure free-field model itself, there is a more direct problem of physical picture.

Excitation is a transitive process.

· An atom transitions from a higher energy level to a lower one, emitting light;
· A charge is accelerated, emitting light;
· A particle and an antiparticle meet and annihilate, emitting light.

The common point is: there needs to be an excitation source and an object being excited.

Light is a product of interaction.

Yet the pure free-field model assumes:

· No excitation source;
· No interaction;
· Only one field, existing by itself.

But it simultaneously requires: this field has zero-point fluctuations and energy.

This is equivalent to requiring:

A system, under conditions of no source and no object, excites itself.

This does not hold as a physical picture.

It can be written as a self-consistent mathematical form, but what this form corresponds to is a loop within the model, not a physical process in nature.

4.3 The Common Structure of Idealized Extrapolation

The two paths of natural elimination presented in this paper hold under the premise of acknowledging the existence of mode coupling and discrete spectral structures in real physical systems. However, some long-standing divergence problems that have troubled researchers in traditional field theory are precisely attached to idealized models that have stripped away these structures. The divergence produced when an idealized model is extrapolated beyond its applicable scope is not an isolated case in the history of science.

Example 1: Point-Charge Self-Energy

Classical electromagnetic theory treats the electron as a geometric point charge. When calculating the electrostatic energy of the point charge itself, letting the distance approach zero causes the integral to diverge.

Mathematical root: assuming that the charge is concentrated at a geometric point with no size.

Example 2: Pure Free-Field Zero-Point Energy

Treating the field as a completely uncoupled free field. Summing the zero-point energies of infinitely many high-frequency modes yields a divergence.

Mathematical root: assuming that there is absolutely no coupling between fields, or between modes.

Example 3: Single-Body Universal Gravitation

Newton's law of universal gravitation

F = G m₁ m₂ / r²

describes the gravitational attraction between two bodies, where r is the distance between them.

If one takes a single celestial body and forcibly applies this two-body formula to itself (setting m₁ = m₂ = M, r = 0), the denominator becomes zero, and the force diverges.

Mathematical root: applying a formula that describes a relation between two bodies to a single object itself.

The common structure of the three examples:

Example Model Extrapolation Direction Result
Point-charge self-energy Point charge Distance → 0 Divergence
Free-field zero-point energy Uncoupled field Mode number → ∞ Divergence
Single-body universal gravitation Single body using two-body formula r = 0 Divergence

None of the three divergences are properties of nature itself; they are results of model extrapolation.

A considerable portion of difficult problems in the history of science arise from this kind of extrapolation. Recognizing whether the model and the object match is more important than solving for the divergence within the model.

V Significance of the Research

The main work of this paper is to present two independent paths of natural elimination of ultraviolet divergence in scenarios involving coupling transitions, while also clarifying the applicable scope of the theory.

Many studies attempt to address the divergence problem of pure free-field zero-point fluctuations, but this problem is attached to an idealized uncoupled model. If the model itself does not correspond to real physics, then the physical value of solving for the divergence in this idealized case needs to be reconsidered.

In the history of science, some long-standing puzzles that have troubled researchers have their roots in the overextension of idealized assumptions. This paper intends to show that it is necessary to distinguish model assumptions from objective physics, and to avoid investing excessive research into idealized models that lack real counterparts.

Pointing out that a question is wrongly posed is itself part of research. This differs from "being unable to solve it"; it points out the premise of the problem, not the ability to compute.

VI Conclusion and Outlook

1. For physical processes involving mode coupling transitions, there are two independent paths of natural elimination:

· Path one: the transition probability decays rapidly as the frequency difference increases, and the loop integral naturally converges;
· Path two: the mode spectrum is discrete, and the number of modes is finite or the frequency has an upper limit, so the summation naturally converges.

2. The two paths can hold independently; if either one is satisfied, the ultraviolet divergence is naturally eliminated.
3. The discrete spectrum is not an external assumption; it comes from any one of finite volume, underlying discrete structure, or interactions.
4. The completely uncoupled pure free field is an idealized model that does not exist in reality; the zero-point fluctuation divergence under this model is not within the scope of the paths addressed in this paper.
5. Future work can further verify the specific form of the amplitude decay, and can also study the combined effects of residual coupling and discrete spectra in real systems.

References

(Field theory and regularization-related literature may be supplemented later)