Proposal for statistical mechanics-based UV regularization using fermion-boson transition functions

Abstract

Introduction:

We propose a statistical‐mechanics–based framework for UV regularization in QED/QFT by introducing energy‐dependent transition functions that interpolate fermionic and bosonic components.

Methods:

We define logistic transition functions T(E) that continuously exchange degrees of freedom between γ_μ and ω_μ operators, and analyze gauge consistency via the Ward–Takahashi identities and BRST symmetry.

Results:

The transition functions act as a smooth, gauge‐safe soft cutoff that exponentially suppresses UV contributions while preserving transversality. We illustrate how longitudinal components are cancelled in internal lines without affecting observables.

Discussion:

This approach offers a physical (statistical) interpretation of regularization, unifies several phenomena across energy scales, and is compatible with Lorentz and gauge symmetries. Extensions to non‐Abelian theories and relations to mass generation mechanisms are outlined.

Rationale:

These points correspond to Supplementary sections S9, S11–S19, S20, etc.

1 Introduction

Quantum field theory (QFT) is the common language of modern physics, with applications ranging from particle physics to condensed matter physics. However, high-order perturbative calculations in QED and QCD face serious mathematical difficulties due to ultraviolet divergences [1–5].

Traditionally, ultraviolet divergences in quantum field theory have been controlled by methods such as cutoffs, dimensional regularization, Wilson’s renormalization group, and renormalization, but these methods rely on formal operations and their physical interpretation is not always self-evident’t [6–8]. In particular, Wilson’s renormalization group provides a powerful framework for explaining scale-dependent effective theories, statistical mechanics phase transitions, and the asymptotic freedom of quantum chromodynamics, but computational complexity and the lack of statistical mechanical perspective remain challenges [8, 9]. For instance, while understanding of the confinement phenomenon in QCD has advanced through lattice gauge theory using the renormalization group, there are limitations in the intuitive description of non-perturbative regions.

This research is a substantially revised and academically reconstructed version of a series of previous publications by the author [10–12, 12]. In this paper, we refer to this framework as Fermion–Boson Duality QED (abbreviated as FBD-QED). This research proposes a new solution to this problem from a statistical mechanical perspective. We introduce the concept of a transition function that depends on energy scale to dynamically change the statistical properties of particles, describing a phenomenon where particles that behave as fermions at low energies transition to bosonic properties at high energy regions, and conversely, photons that behave as bosons at low energies exhibit fermionic properties at high energies. This concept of statistical phase transition aligns with recent trends attempting to explain diverse physical systems by extending the Fermi-Dirac distribution.

Originally proposed as a model for electron gas, the Fermi-Dirac distribution has been observed and utilized in various environments, including analog gravity systems using water waves [14], non-Hermitian mesoscopic rings [15], and semiconductor devices [16]. This research extends the concept of “environment-dependent deformed distribution functions” to high-energy physics, exploring applications not only for ultraviolet divergences in QED but also for non-abelian gauge theories like QCD.

Conventionally, fermions (like electrons) and bosons (like photons) have been considered distinct particles with exclusive statistics. However, this research examines the possibility that statistical properties may change dynamically depending on energy scales. Specifically, we assume that electrons, which behave as fermions at low energies, exhibit bosonic behavior at high energies, and conversely, photons undergo a dual transition to fermionic aspects.

When transition functions are incorporated into QED amplitude calculations, contributions from the ultraviolet region naturally attenuate, suppressing divergences. Using the electron self-energy as a concrete example, we numerically evaluate how the introduction of transition functions converges divergent integrals to finite values. This approach may open a path to physically regularizing QFT without introducing arbitrary cutoffs or renormalization constants.

From a statistical mechanical perspective, it is not uncommon for the macroscopic behavior of particle ensembles to undergo qualitative changes due to energy. In Cooper pair formation in superconductivity, electrons, which are fermions, effectively become bosonized and condense [17, 18]. Statistical properties are also known to be modified by thermal corrections in finite temperature field theory. This research extends these analogies to extremely high energies approaching the Planck scale, examining scenarios where particle statistics themselves are transformed.

In Section 2 we explain in detail the duality and the transition functions, while Section 3 constructs the extended QED. Section 4 demonstrates the effectiveness of the method through an explicit calculation of the electron self-energy, and Section 5 concludes by summarizing the significance of this work and the remaining open problems. A more detailed mathematical and physical justification of our approach is provided in the Supplementary Material; a concise overview is given in Supplementary Material. The Supplementary Material discusses, in depth, the validity of the two-dimensional Lorentz transformation, the physical basis of spin–statistics separation, the interpretation of the bosonic tensor as an energy–momentum tensor, the consistency between the Ward–Takahashi identities and the transition-function formalism, the transverse wave projector and gauge symmetry, compatibility with BRST transformations, potential applications to QCD and other theories, and its relationship to the Higgs mechanism. Section 6 describes the Mathematica code used for the numerical calculations.

These results demonstrate that the transition function framework provides a robust theoretical foundation that suppresses ultraviolet divergences while preserving gauge symmetry.

This research is positioned at the intersection of QFT and statistical mechanics, approaching mathematical challenges in high-energy physics through the new perspective of energy scale-dependent statistical transitions. This viewpoint is expected to have ripple effects on phase transition research in complex systems, deepening understanding of “statistical transitions” as universal phenomena transcending material hierarchy.

2 Theoretical framework of fermion-boson duality

2.1 A new understanding of statistical properties: “Separation” of spin and statistics

One of the fundamental principles of quantum mechanics is the spin-statistics theorem, which connects a particle’s spin with its statistical nature. According to this theorem, particles with half-integer spin (e.g., , ) are fermions, and particles with integer spin (e.g., 0, 1, 2) are bosons [19]. This relationship has long been accepted as a fundamental framework in elementary particle physics.

This framework relaxes the conventional constraint that spin and statistics must strictly follow different representations of the Lorentz group, modeling energy-dependent changes in statistics as an effective theory approach. For example, in superconductivity, spin electrons effectively demonstrate bosonic behavior in Cooper pair formation (see Supplementary MaterialSection 2). Similarly, we assume that electrons can transition to an effective spin one bosonic state inside atoms or in high-energy regions.

In this theoretical framework, spin and statistics are treated as independent characteristics that can change depending on energy scales and physical conditions. In the low-energy limit, electrons behave as fermionic electrons and photons as bosonic photons, consistent with conventional quantum field theory. However, in the high-energy limit, electrons may transition to bosonic electrons and photons to fermionic photons.

To represent these states, we define the total state vector of the system as in Equation 1.

The visualization of this state is shown in Figure 1.

FIGURE 1

20, 21] requires new particles and higher-dimensional spaces, FBD-QED models statistical transition within the same particle inspired by semiconductor theory.

Table 1 shows correspondence examples of the four elementary particle states.

The complete quantum state of each particle is expressed as an energy-dependent linear combination of these basis states:

Here, represents the transition function that determines the weight of each statistical component at a specific energy scale .

2.1.1 Reality of “bosonic electrons” and “fermionic photons” is understood as effective hybrid states

     The Coulomb field around atomic nuclei gives electrons an effective pressure equivalent to , locally forming a “room temperature, ultra-high pressure” superconducting state comparable to Cooper pair condensation at extremely low temperatures.

     In these extreme environments, electrons (spin ) retain their fermionic intrinsic spin while acquiring bosonic correlation components through interactions with the photon field. Specifically, the transition functions and are simultaneously non-zero and satisfy , so electrons behave as effective quasiparticles exhibiting fermion–boson duality. Similarly, photons can also take hybrid states with .

     Since the fermionic component of the transition function always remains, single-electron operators satisfy anticommutation relations, and unlimited condensation into the 1s orbital does not occur. Furthermore, when is non-zero, the fermionic exclusion effect on the photon side also works as statistical complementarity to suppress excessive electron occupation.

     When electrons or photons escape from atoms, the ultra-high pressure environment is instantly lost, and like ice melting into water in an instant, the statistics immediately return to their standard forms (fermionic electrons/bosonic photons). Therefore, detectors only detect normal electrons and photons.

Therefore, our model does not claim that electrons become pure bosons inside atoms, but rather that they behave as hybrid quasiparticles with finite fermionic components, thus not destroying the structure of the periodic table or chemical bonding.

2.2 Introduction of transition functions and their physical meaning

Transition functions are mathematical tools that quantify the transition of a particle’s statistical properties accompanying energy changes, defined as follows:

It is notable that has a form similar to the Fermi distribution function. In this research, transition functions are interpreted as quantum mechanical extensions of the Fermi distribution function in single elementary particle systems. This suggests a deep connection between statistical properties and thermal statistical mechanics.

Transition functions satisfy the following conservation laws:

These equations show that the sum of fermionic and bosonic components within the same particle species is always 1, describing the transition of statistical properties with energy changes in a consistent manner.

Figure 2 conceptually shows the four-quadrant representation of transition functions. Furthermore, Figure 3 demonstrates the continuous redistribution of four-component probabilities during actual energy sweeping. Here, the curve does not represent “the Bose-Einstein distribution itself,” but rather depicts the probability weight for to maintain bosonic properties. Therefore, thermal equilibrium features such as divergence in the low-energy limit or zero-mode condensation do not appear in this figure.

FIGURE 2

FIGURE 3

2.3 Relationship between transition functions and the Hill–Wheeler equation

The transition function takes the form of a logistic function see Equation 4:

It has the same form as the normal Fermi–Dirac distribution . This function form is known as the Hill–Wheeler equation [22–24] in nuclear fission theory, which was originally a formula for calculating the transmission probability when treating nuclear fission barriers using harmonic oscillator approximation [25–31]. This research reinterprets it not merely as a nuclear fission transmission coefficient, but as a quantum statistical occupation probability [32].

This perspective naturally explains changes in statistical properties in high-energy regions, and the utility of this interpretation is demonstrated in the numerical analysis discussed later.

2.4 Correspondence with semiconductor physics

FIGURE 4

Using this approach, it may be possible to construct equations that avoid infinities in vacuum polarization, electron self-energy, and vertex corrections without artificial regularization.

Note that the state referred to as the “intermediate region” in this paper is a mixed statistics where fermionic component and bosonic component coexist probabilistically, which is a different concept from anyons (limited to two dimensions) [33]; [34]; [35] that have continuously variable particle exchange phases.

2.5 Boundary conditions and region characteristics of transition functions

2.5.1 Theoretical background and positioning

The division of degrees of freedom according to energy scale has been discussed for a long time in (i) BCS theory [36]; [17] where low-temperature Fermi systems exhibit boson condensation behavior, (ii) the rapid change of effective degrees of freedom near transition scales shown by Wilson’s successive integration-type renormalization group [9] and Miransky scaling [37]; [38], and (iii) asymptotic freedom in QCD [39]; [40]. The novelty of this research lies in extending the concept of these “multi-scale effective theories” to energy-dependent transitions of spin and statistics, constructing the theory based on the following three regions:

2.5.2 Specific form of transition functions

The transition functions used in this research areSince the transition functions in Equation 7 are logistic, holds identically. The parameter settings are summarized in Table 2.

The dimensionless procedure and physical unit restoration method are detailed in Section 4.5.

Implementation results are detailed in Section 4.5 and Section 6 [Zenodo DOI 10.5281/zenodo.15825707 (Version 4)].

2.5.3 Universality and model dependence

The mathematical form (S-curve shape) of transition functions is universal, but the specific numerical values vary greatly depending on physical situations. This is similar to how “Fermi distribution functions have the same S-shape for both electrons and holes, but the temperature and chemical potential values differ for each material.”

This shows a hierarchical structure, suggesting that different statistical transitions may occur at different energy scales.

Understanding through familiar examples: This is similar to how “water’s boiling point changes with atmospheric pressure (87°C on Mount Fuji), but the boiling phenomenon itself (liquidgas phase transition) is universal.” The statistical transition phenomenon is universal, but the energy at which it occurs and its sharpness depend on the environment (theoretical framework).

2.6 Realization examples in condensed matter physics

These examples support the concept that changes in statistical properties dependent on energy scales, as proposed in this theory, are applicable not only to high-energy physics but also to condensed matter physics.

3 Bosonic gamma matrices and extended quantum electrodynamics

To incorporate the concept of fermion-boson duality introduced in the previous section into the framework of quantum field theory, an extension of the conventional Dirac equation is necessary. In this section, we introduce bosonic gamma matrices to realize this extension and construct an extended quantum electrodynamics Lagrangian based on them.

3.1 Introduction of bosonic gamma matrices

In this research, to describe the transformation of statistics from fermions to bosons, we introduce new bosonic gamma matrices . These matrices are defined using the conventional Dirac matrices and another representation that follows the same Clifford algebra

Here, , as introduced in Equation 8, is defined by the following substitutions:

This specific substitution can be represented using an appropriate unitary transformation as , satisfying the Clifford algebra . This construction is similar to the process of fermion electrons forming Cooper pairs in superconductivity, but differs in that it reconstructs the internal degrees of freedom of a single particle to realize a statistics transformation rather than combining two particles. To use an analogy, acts as a transformation operator when an electron “wears bosonic clothes,” functioning as a “magic scissors” that restricts particle motion to two transverse components.

With this definition, the bosonic gamma matrices corresponding to a particle directed along the -axis are expressed as:

These bosonic gamma matrices satisfy the following important anticommutation relations:

Here, represent spinor indices, and is the identity matrix. This algebraic structure gives rise to “semi-Hermitian components” and “semi-anti-Hermitian components” different from conventional , yielding operators with different properties in the time and space directions.

The explicit matrix representation of bosonic gamma matrices is given by:

This matrix structure enables the description of particles incorporating both bosonic and fermionic properties in the extended Lagrangian shown in Section 3.2.

For details of this calculation, please refer to the Mathematica code and calculation results available from the Zenodo repository provided in Section 6.

3.2 Lagrangian of extended quantum electrodynamics

3.2.1 Four basis states

The fermion/boson four components of “electron (e)” and “photon ” introduced in the previous section arewhere , represent the fermion/boson components of electrons, and , represent the fermion/boson components of photons, as defined in Equation 13.

3.2.2 Definition of transition functions

Scalar functions that depend on energy and satisfyare called “transition functions.” In the low-energy limit , , , recovering standard QED, and at high energies, both approach 0, making the statistical phase transition manifest.

3.2.3 Extended Lagrangian

With the ordinary Dirac matrices , the bosonic gamma matrices , and the covariant derivative , the minimal Lagrangian of the extended QED is given by Equation 14:where is the usual electromagnetic field-strength tensor, and is the corresponding fermionic tensor (see Supplementary Material, Section 3, for its definition).

3.2.4 Electron (fermion) kinetic term:

3.2.5 Electron (boson) kinetic term:

3.2.6 Boson field kinetic term:

3.2.7 Fermion field kinetic term

3.2.7.1 Behavior in low and high energy limits

3.2.7.2 Preservation of gauge invariance

Under the usual transformations for the electron field and for the gauge potential, the extended Lagrangian remains invariant. Because the transition functions are introduced as Lorentz scalars, , gauge symmetry—including the BRST transformations [42, 43]—is preserved (see Supplementary Material, Section 5, for details). Furthermore, Supplementary Material, Section 4, verifies that the introduction of transition functions leaves the Ward–Takahashi identities [44, 45] intact, so the symmetry between vertex functions and full propagators continues to hold.

3.2.7.3 Physical implications

3.3 Theoretical basis for bosonic kinetic term

     In the same spirit that Dirac’s equation “elevated the second-derivative Schrödinger equation to first-derivative to succinctly describe relativistic fermions,” this research rewrites the second-derivative Klein-Gordon equation [4, 52–56] in first-derivative form, unifying fermions and bosons witha single first-order operator (as defined in Equation 15).

Viewing Equation 16 as , the conjugate momentum for time derivatives isAs shown in Equation 17, the coefficients are independent of . Therefore, the formula for LagrangianHamiltonian Legendre transformation can be completed in one line regardless of particle type.

Similarly, the Fourier transform of the equation of motion is , and the propagator (Green’s function) is given by Equation 18obtained with a one-pattern inverse matrix. If then , if then , changing automatically, eliminating the need to distinguish “with/without gamma matrices” when performing loop calculations.

This is like consolidating separate “tools” for electrons (fermions) and photons (bosons) into a single universal wrench, greatly simplifying theoretical calculations.

In essence, is “a screwdriver with only two handles from the start,” with no superfluous contact points with screw holes (physical states). When quantizing fields, the three-step ritual of “gauge fixing constraint conditions physical state selection” becomes entirely unnecessary, making both theory construction and practical calculations instantly simpler.

4 Automatic avoidance of ultraviolet divergence—natural regularization by transition functions

In this section, we demonstrate how the transition functions (fermionic degree of electrons) and (bosonic degree of photons) can be inserted into the three types of one-loop integrals—vacuum polarization, electron self-energy, and vertex correction—to suppress ultraviolet divergence. In each example, we can observe a consistent mechanism that “reproduces standard QED at low energies and introduces exponential decay at high energies.”

4.1 Vacuum polarization

The vacuum polarization tensor in standard QED is given by Equation 20 [1, 4, 52–54, 62–66]:which has a divergence proportional to in the high-energy region. Modifying this with the substitution of transition functions for electron lines yields the expression in Equation 21As , the exponential factor appears twice, making the integral kernel decay rapidly and eliminating the divergence.

4.2 Electron self-energy

In a scalar toy model (omitting spinor traces), the standard self-energy is given by Equation 22 [4, 52–54, 62–64, 67–69]:Inserting transition functions for both electrons and photons gives Equation 23As , and , causing the integral kernel to be suppressed by , making converge to a finite value (In the numerical example of Sec. 4.5, is suppressed to ).

4.3 Vertex correction

One-loop vertex function (scalar approximation) [4, 44, 52–54, 62–65, 70] is given by Equation 24:With similar substitutions, we obtain Equation 25:

In the limit, the presence of the transition functions with a cubic power leads to an exponential suppression that is even faster than the standard fall-off, ensuring convergence to a finite value. The Ward–Takahashi identity remains valid after the inclusion of the transition functions (see Supplementary Material, Section 4, for the proof).

4.4 Specific form of transition functions

At , , reproducing standard QED, and at , both fall exponentially to 0, suppressing divergent terms (see Equation 26).

Summary: For vacuum polarization, self-energy, and vertex correction, all integrals have “transition functions cubed or less” as exponential decay factors, automatically converging without introducing cutoffs or renormalization constants. This is the core result of “statistical regularization.”

In other words, whereas the conventional theory predicts a divergence as the separation approaches zero, the model proposed herein suppresses this singularity to a finite value through the transition mechanism. This idea is illustrated in Figure 5.

FIGURE 5

4.5 Numerical calculation example of electron self-energy correction and natural regularization by transition functions

This calculation verifies the effect of transition function in suppressing divergence in high-energy regions using electron self-energy correction in quantum electrodynamics (QED) as a subject. We adopt a scalar toy model that omits spinor structure and gauge fixing in rigorous QED calculations to demonstrate the principles of transition functions. Details are provided in Appendix6. The simplified self-energy equation presented in Section 4.2 provides the conceptual foundation for this toy model, but here we explain the transition from rigorous QED equations to the toy model.

4.5.1 Numerical calculation as a toy model and its results

Electron self-energy is a typical example of QED one-loop corrections that diverges as global momentum . In this section, we introduce fermion-boson transition function and quantitatively demonstrate how divergence is exponentially suppressed using a simplified model.

4.5.1.1 Exact expression and simplification

reducing to the scalar toy model in Equation 28:With , , , we getreproducing divergent growth (see Equation 29).

4.5.1.2 Introduction of transition functions

Modeling statistical phase transition withand applying it to internal lines of both electrons and photons gives Equation 30With the same parameters, numerical integration yields Equation 31showing orders of magnitude reduction compared to the no-transition case, demonstrating a pronounced exponential cutoff effect.

Note: The numerical values in this section are rough estimates from deterministic one-dimensional integration toy models, and statistical confidence intervals (-fit or Monte Carlo errors) have not been evaluated.

4.5.1.3 Parameter sensitivity

Results with varying transition width are shown in Table 3. Smaller widths make the logistic function steeper, strengthening suppression, which can be quantitatively confirmed.

4.5.1.4 Theoretical consistency

Thus, transition functions have been numerically verified to play the role of “naturally” cutting off ultraviolet divergence.

4.5.2 Physical interpretation of transition function parameters

4.5.2.1 Discussion

4.5.2.1.1 Physical Insights and Condensed Matter Analogies.

**Physical Consequences:** - Divergence suppression also becomes gradual, with intermediate statistical mixed states existing widely - The region where fermionic and bosonic components coexist expands - Example: Phenomena where photons partially exhibit fermionic properties become more observable.

**Scattering Experiments:** Using experimental apparatus with good energy resolution to precisely measure the energy dependence of scattering cross-sections, estimating from the “steepness of the slope” of the S-curve.

**Semiconductor Analogy:** The same statistical analysis used in semiconductors to measure changes in electron concentration while varying temperature Street [74] can be applied. Energy replaces temperature, and statistical component ratios replace electron concentration.

Key points: Small : Abrupt transition, clear threshold effects - Large : Smooth transition, coexistence phenomena - Both correspond deeply to phase transition phenomena in nature.

From the above interpretation, it is understood that represents the typical scale at which statistics transition, and represents the width (sharpness) of the transition. Therefore, if these two parameters can be determined through experimental or numerical approaches, it becomes possible to quantitatively grasp “at which energy region and with what sharpness statistical properties switch”. This is a major feature of this theory and is key to explaining the suppression mechanism of divergence in high-energy regions in a form different from conventional renormalization methods.

4.5.2.2 Practical procedures for parameter extraction

In actual physics, analogies with semiconductor bandgap measurements would be useful. In semiconductors, established techniques exist for determining the central energy of bandgaps (corresponding to ) and transition steepness (corresponding to ) by measuring changes in electron concentration while varying temperature and analyzing the resulting “S-curves.” This suggests that similar statistical analysis methods could be applied to our theory.

4.5.2.3 Physical image of convergence by statistical transition

This research’s fermion-boson duality theory naturally suppresses divergence in high-energy regions through energy-dependent transition of particle statistics.

Figure 4 shows the energy band diagram, density of states, and distribution function of fermionic electrons. At the Fermi energy , the existence probability of fermionic electrons decreases sharply, with some transitioning to bosonic electrons. This transition stabilizes the electron distribution and suppresses excessive contributions at high energies. This numerical calculation quantitatively supports this stabilization.

Figure 5 compares conventional QED models with this framework. In conventional models, interactions of fermionic electrons (eF) or bosonic photons diverge as (high-energy region), but in this framework, as , fermionic electrons transition to bosonic electrons (eB) and bosonic photons to fermionic photons , causing interactions to converge finitely. This statistical transition is realized through the logistic form of the transition function , exponentially suppressing existence probability at high energies.

Conventional renormalization theory derives effective physical quantities through infinite-infinite subtraction, ignoring the physical reality that existence probability at high energies follows , approaching zero. In this framework, the logistic form of transition functions naturally suppresses divergence, avoiding the artificiality of renormalization. This natural regularization is intuitively understood through the statistical transition in Figures 4, 5, emphasizing the novelty of this theory.

4.5.2.4 Scale setting and dimensionless analysis

The numerical examples in this section (1) adopt natural units , and (2) calculate after dimensionless normalization by dividing all momenta and masses by the transition threshold energy . Specifically, as defined in Equation 32:The integration variable is replaced by , and to restore the dimension of self-energy, we use . Therefore, the notation in the numerical examples represents the dimensionless coefficient with as the reference unit. The actual physical values can be easily rescaled according to the choice of (e.g., 10 MeV).

4.5.2.5 Quantitative relationship between transition width and UV suppression

Since the high-momentum asymptotic behavior of the transition function is , the one-loop self-energy integral is approximately given by Equation 33That isAs shown in Equation 34, also appears in the denominator of the exponential decay factor. Therefore, smaller(steep transition) results in stronger exponential cutoff, significantly suppressing UV divergence. In the numerical implementation of Section 4.5, merely halving from reduces the self-energy by approximately times (see Table 3). This sharpening/softening corresponds to condensed matter analogies such as gap opening/closing rates in BCS–BEC crossovers or metal-insulator transitions Imada et al. [72]; Chen et al. [41].

5 Conclusions and outlook

In this work we introduced a transition function, by which particle statistics change continuously from Fermi–Dirac to Bose–Einstein as the energy increases. On this basis we proposed a statistical regularization scheme that treats ultraviolet (UV) divergences and mass generation in quantum electrodynamics (QED) within a single, unified framework. Below we summarize the main achievements and outline future tasks together with the broader perspective opened by the present approach.

5.1 Key achievements

     As shown in Supplementary Material, Section 6, extending the same transition function to quarks and gluons yields an Lagrangian, producing a qualitative scenario in which the gluon mass and color confinement are explained by a single logistic transition.

5.2 Future prospects

Higher-order calculations (two loops and beyond) of the transition-function function, together with comparisons to lattice QED/QCD simulations, will quantify the universality of the UV-suppression effect.

By fitting the threshold and width to hadron masses and scattering data, the experimental scale of the transition parameters can be determined.

A unified treatment of thermodynamic and geometric entropy may connect this framework to black-hole evaporation and early- Universe inflation.

By comparing fermionboson conversions in superconducting gap formation and exciton condensation, the transition concept could be systematized as a cross-disciplinary theme spanning condensed-matter and high-energy physics.

The toy model in this paper only treated deterministic integrals. In the future, we plan to use Monte Carlo integration and Bayesian error propagation to estimate posterior distributions of parameters, and compare them with multi-loop and lattice calculations to demonstrate divergence suppression quantities with error bounds.

5.3 Summary

The transition-function framework reinterprets the traditional “mathematical tricks” of regularization and renormalization as a statistical-mechanical process, thereby opening a new path for simultaneous control of UV divergences, mass generation, and gauge symmetry. The results presented here constitute a conceptual blueprint, and a rich program of higher-order theory, numerical implementation, and experimental confrontation is expected to promote a multifaceted bridge between high-energy and statistical physics.

6 Attached mathematica programs

Below is a brief explanation of the calculation content of the two MATHEMATICA programs included in the repository.

6.1 ElectronSelfEnergy_Regularization.nb

This demonstrates that the introduction of transition functions suppresses contributions from high momentum regions, yielding finite values without renormalization.

6.2 omega_matrix_properties.nb

This program defines standard 4 four gamma matrices and confirms their properties. It is used to verify two-dimensional Lorentz transformations in the extended QED of this research. Specifically, it explicitly describes and attempts to confirm anticommutation relations by calculating products such as . This provides the foundation for the possibility of introducing bosonic gamma matrices .

6.3 TransitionFunction_Visualizer.nb

This notebook is a visualization tool that generates probability distributions of the four components based on the logistic transition function as (1) interactive manipulation, (2) static snapshots, (3) GIF/MP4 animations. The default values clearly reproduce at low energies and at high energies.

References

• 1.

  KawamuraY. Elementary particle physics. Tokyo, Japan: Shokabo Publishing (2024).

  • Google Scholar
  • View reference in article

• 2.

  PeskinMSchroederD. An introduction to quantum field theory. Reading, MA, USA: Addison-Wesley (1995).

  • Google Scholar
  • View reference in article

• 3.

  WeinbergS. The quantum theory of fields, vols. I–II. Cambridge, UK: Cambridge University Press (1995).

  • Google Scholar
  • View reference in article

• 4.

  KawamuraY. Relativistic quantum mechanics. Tokyo, Japan: Shokabo (2012).

  • Google Scholar
  • View reference in article

• 5.

  SakamotoM. Quantum field theory (II). Tokyo, Japan: Shokabo (2020).

  • Google Scholar
  • View reference in article

• 6.

  HooftGVeltmanM. Diagrammar. Tech Rep CERN Yellow Rep (1973) 73–9. 10.5170/CERN-1973-009

  • CrossRef
  • Google Scholar
  • View reference in article

• 7.

  BogoliubovNShirkovD. Introduction to the theory of quantized fields. New York, NY, USA: Interscience (1959).

  • Google Scholar
  • View reference in article

• 8.

  WilsonK. The renormalization group: critical phenomena and the kondo problem. Rev Mod Phys (1975) 47:773–840. 10.1103/revmodphys.47.773

  • CrossRef
  • Google Scholar
  • View reference in article

• 9.

  WilsonK. Confinement of quarks. Phys Rev D (1974) 10:2445–59. 10.1103/physrevd.10.2445

  • CrossRef
  • Google Scholar
  • View reference in article

• 10.

  ClubMD. Strange equations in elementary particle theory: fermion-boson duality type quantum electrodynamics (2020). Available online at: https://www.amazon.co.jp/dp/B086SCJL3T.

  • Google Scholar
  • View reference in article

• 11.

  ClubMD. Strange equations in elementary particle theory: fermion-boson duality type quantum chromodynamics (2020). Available online at: https://www.amazon.co.jp/dp/B08NT3KCNC.

  • Google Scholar
  • View reference in article

• 12.

  ClubMD. On fermion/boson dual quantum electrodynamics: strange mathematical formulas in elementary particle theory. Seattle, WA, United States: Kindle Direct Publishing (Amazon) (2024). Available online at: https://www.amazon.com/dp/B0CXN36G66.

  • Google Scholar
  • View reference in article

• 13.

  ClubMD. Quantum chromodynamics with fermion-boson duality: strange equations in particle theory. Seattle, WA, United States: Kindle Direct Publishing (Amazon) (2024). Available online at: https://www.amazon.com/dp/B0DRD939DM.

  • Google Scholar
  • View reference in article

• 14.

  RozenmanGUllingerFZimmermannMEfremovMAShemerLSchleichWPet alObservation of a phase-space horizon with surface-gravity water waves. Commun Phys (2024) 7:165. 10.1038/s42005-024-01616-7

  • CrossRef
  • Google Scholar
  • View reference in article

• 15.

  ShenP-XLuZLadoJTrifM. Non-hermitian fermi–dirac distribution in persistent-current transport. Phys Rev Lett (2024) 132:086301. 10.1103/physrevlett.133.086301

  • Pubmed Abstract
  • CrossRef
  • Google Scholar
  • View reference in article

• 16.

  SueyasuT. Introduction to optical devices. Tokyo, Japan: Corona Publishing (2018).

  • Google Scholar
  • View reference in article

• 17.

  BardeenJCooperLSchriefferJ. Theory of superconductivity. Phys Rev (1957) 108:1175–204. 10.1103/physrev.108.1175

  • CrossRef
  • Google Scholar
  • View reference in article

• 18.

  GinzburgVLandauL. On the theory of superconductivity. Zh Eksp Teor Fiz (1950) 20:1064–82.

  • Google Scholar
  • View reference in article

• 19.

  PauliW. The connection between spin and statistics. Phys Rev (1940) 58:716–22. 10.1103/physrev.58.716

  • CrossRef
  • Google Scholar
  • View reference in article

• 20.

  NillesH. Supersymmetry, supergravity and particle physics. Phys Rep (1984) 110:1–162. 10.1016/0370-1573(84)90008-5

  • CrossRef
  • Google Scholar
  • View reference in article

• 21.

  HaberHKaneG. The search for supersymmetry: probing physics beyond the standard model. Phys Rep (1985) 117:75–263. 10.1016/0370-1573(85)90051-1

  • CrossRef
  • Google Scholar
  • View reference in article

• 22.

  HillDWheelerJ. Nuclear constitution and the interpretation of fission phenomena. Phys Rev (1953) 89:1102–45. 10.1103/physrev.89.1102

  • CrossRef
  • Google Scholar
  • View reference in article

• 23.

  LandauLLifshitzE. Quantum mechanics: non-relativistic theory. 3rd ed., 3. Oxford, UK: Pergamon Press (1977).

  • Google Scholar
  • View reference in article

• 24.

  RoyRNigamB. Nuclear physics: theory and experiment. New York, NY, USA: John Wiley and Sons (1967).

  • Google Scholar
  • View reference in article

• 25.

  RagnarssonINilssonS. Shapes and shells in nuclear structure. Cambridge, UK: Cambridge University Press (1995).

  • Google Scholar
  • View reference in article

• 26.

  TakahashiAOhtaMMizunoT. Production of stable isotopes by selective-channel photofission of pd. Jpn J Appl Phys (2001) 40:7031–4. 10.1143/jjap.40.7031

  • CrossRef
  • Google Scholar
  • View reference in article

• 27.

  OhtaMMatsunakaMTakahashiA. Analysis of²³⁵u fission by selective-channel scission model. Jpn J Appl Phys (2001) 40:7047–51. 10.1143/jjap.40.7047

  • CrossRef
  • Google Scholar
  • View reference in article

• 28.

  OhtaMTakahashiA. Energy-dependence of fission-product yields for²³⁵u. Jpn J Appl Phys (2003) 42:645–9. 10.1143/JJAP.42.6445

  • CrossRef
  • Google Scholar
  • View reference in article

• 29.

  OhtaMNakamuraS. Channel-dependent fission barriers of n+²³⁵u. Jpn J Appl Phys (2006) 45:6431–6. 10.1143/jjap.45.6431

  • CrossRef
  • Google Scholar
  • View reference in article

• 30.

  OhtaMNakamuraS. Simple estimation of fission yields with selective-channel scission model. J Nucl Sci Technol (2007) 44:1491–9. 10.3327/jnst.44.1491

  • CrossRef
  • Google Scholar
  • View reference in article

• 31.

  OhtaM. Influence of deformation on fission yield in selective-channel scission model. J Nucl Sci Technol (2009) 46:6–11. 10.3327/jnst.46.6

  • CrossRef
  • Google Scholar
  • View reference in article

• 32.

  MaruyamaH. Application of the hill-wheeler formula in statistical models of nuclear fission: a statistical–mechanical approach based on similarities with semiconductor physics. Entropy (2025) 27:227. 10.3390/e27030227

  • Pubmed Abstract
  • CrossRef
  • Google Scholar
  • View reference in article

• 33.

  SternA. Anyons and the quantum hall effect: a pedagogical review. Ann Phys (2008) 323:204–49. 10.1016/j.aop.2007.10.008

  • CrossRef
  • Google Scholar
  • View reference in article

• 34.

  WilczekF. Fractional statistics and anyon superconductivity. World Scientific Monograph/book (1990).

  • Google Scholar
  • View reference in article

• 35.

  JainJ. Composite fermions. Cambridge, UK: Cambridge University Press (2007). 10.1017/CBO9780511618952

  • CrossRef
  • Google Scholar
  • View reference in article

• 36.

  CooperL. Bound electron pairs in a degenerate fermi gas. Phys Rev (1956) 104:1189–90. 10.1103/PhysRev.104.1189

  • CrossRef
  • Google Scholar
  • View reference in article

• 37.

  MiranskyV. Dynamics of spontaneous chiral symmetry breaking and continuum limit in quantum electrodynamics. Nucl Phys B (1984) 235:149–76. 10.1016/0550-3213(84)90164-3

  • CrossRef
  • Google Scholar
  • View reference in article

• 38.

  YamawakiK (1996). Dynamical symmetry breaking with large anomalous dimension.

  • Google Scholar
  • View reference in article

• 39.

  GrossDWilczekF. Ultraviolet behavior of non-abelian gauge theories. Phys Rev Lett (1973) 30:1343–6. 10.1103/PhysRevLett.30.1343

  • CrossRef
  • Google Scholar
  • View reference in article

• 40.

  PolitzerH. Reliable perturbative results for strong interactions?Phys Rev Lett (1973) 30:1346–9. 10.1103/physrevlett.30.1346

  • CrossRef
  • Google Scholar
  • View reference in article

• 41.

  ChenQStajicJTanSLevinK. Bcs–bec crossover from high-t_c superconductors to ultracold superfluids. Phys Rep (2005) 412:1–88. 10.1016/j.physrep.2005.02.005

  • CrossRef
  • Google Scholar
  • View reference in article

• 42.

  BecchiCRouetAStoraR. Renormalization of gauge theories. Ann Phys (1976) 98:287–321. 10.1016/0003-4916(76)90156-1

  • CrossRef
  • Google Scholar
  • View reference in article

• 43.

  TyutinI. Gauge invariance in field theory and statistical physics in operator formalism. Lebedev Inst preprint (1975) 39. 10.48550/arXiv.0812.0580

  • CrossRef
  • Google Scholar
  • View reference in article

• 44.

  WardJ. An identity in quantum electrodynamics. Phys Rev (1950) 78:182. 10.1103/physrev.78.182

  • CrossRef
  • Google Scholar
  • View reference in article

• 45.

  TakahashiY. On the generalized ward identity. Nuovo Cim (1957) 6:371–5. 10.1007/bf02832514

  • CrossRef
  • Google Scholar
  • View reference in article

• 46.

  ProcaA. Sur la théorie ondulatoire des électrons positifs et négatifs. J Phys Radium (1936) 7:347–53.

  • Google Scholar
  • View reference in article

• 47.

  AiharaH. Physics of elementary particles. Tokyo, Japan: University of Tokyo Press (2006).

  • Google Scholar
  • View reference in article

• 48.

  HiokiY. Relativistic quantum fields. Kyoto, Japan: Yoshioka Shoten (2008).

  • Google Scholar
  • View reference in article

• 49.

  WessJZuminoB. Supergauge transformations in four dimensions. Nucl Phys B (1974) 70:39–50. 10.1016/0550-3213(74)90355-1

  • CrossRef
  • Google Scholar
  • View reference in article

• 50.

  WessJBaggerJ. Supersymmetry and supergravity. 2nd ed.Princeton, NJ, USA: Princeton University Press (1992).

  • Google Scholar
  • View reference in article

• 51.

  DiracP. The quantum theory of the electron. Proc R Soc Lond A (1928) 117:610–24. 10.1098/rspa.1928.0023

  • CrossRef
  • Google Scholar
  • View reference in article

• 52.

  MandlFShawG. Quantum field theory. 2nd ed. New York, NY, USA: John Wiley and Sons (1984).

  • Google Scholar
  • View reference in article

• 53.

  HalzenFMartinA. Quarks and leptons. New York, NY, USA: John Wiley and Sons (1984).

  • Google Scholar
  • View reference in article

• 54.

  AitchisonIHeyA. Gauge theories in particle physics. 4th ed.Boca Raton, FL, USA: CRC Press (2013).

  • Google Scholar
  • View reference in article

• 55.

  KleinO. Quantentheorie und fünfdimensionale relativitätstheorie. Z Phys (1926) 37:895–906. 10.1007/bf01397481

  • CrossRef
  • Google Scholar
  • View reference in article

• 56.

  GordonW. Der comptoneffekt nach der schrödingerschen theorie. Z Phys (1926) 40:117–33.

  • Google Scholar
  • View reference in article

• 57.

  KugoT. Quantum theory of gauge fields I. New Phys Ser (1989) 23.

  • Google Scholar
  • View reference in article

• 58.

  HiokiY. Quantum field theory: fundamentals of perturbation calculations (kyoto, Japan: yoshioka shoten). 3rd ed. (2022). (in Japanese).

  • Google Scholar
  • View reference in article

• 59.

  GuptaS. Theory of longitudinal photons in quantum electrodynamics. Proc Phys Soc A (1950) 63:681–91. 10.1088/0370-1298/63/7/301

  • CrossRef
  • Google Scholar
  • View reference in article

• 60.

  BleulerK. Eine neue methode zur behandlung der longitudinalen und skalaren photonen. Helv Phys Acta (1950) 23:567–86.

  • Google Scholar
  • View reference in article

• 61.

  FaddeevLPopovV. Feynman diagrams for the yang–mills field. Phys Lett B (1967) 25:29–30. 10.1016/0370-2693(67)90067-6

  • CrossRef
  • Google Scholar
  • View reference in article

• 62.

  SakuraiJ. Advanced quantum mechanics. Reading, MA, USA: Addison-Wesley (1967).

  • Google Scholar
  • View reference in article

• 63.

  LandauLLifshitzE. Relativistic quantum theory. Oxford, UK: Pergamon Press (1971).

  • Google Scholar
  • View reference in article

• 64.

  SakuraiJNapolitanoJ. Modern quantum mechanics. 2nd ed.Cambridge, UK: Cambridge University Press (2017).

  • Google Scholar
  • View reference in article

• 65.

  SchwingerJ. On quantum electrodynamics and the magnetic moment of the electron. Phys Rev (1948) 73:416–7. 10.1103/physrev.73.416

  • CrossRef
  • Google Scholar
  • View reference in article

• 66.

  UehlingE. Polarization effects in the positron theory. Phys Rev (1935) 48:55–63. 10.1103/physrev.48.55

  • CrossRef
  • Google Scholar
  • View reference in article

• 67.

  DysonF. The radiation theories of tomonaga, schwinger, and feynman. Phys Rev (1949) 75:486–502. 10.1103/physrev.75.486

  • CrossRef
  • Google Scholar
  • View reference in article

• 68.

  DysonF. The s-matrix in quantum electrodynamics. Phys Rev (1949) 76:1736–55. 10.1103/physrev.75.1736

  • CrossRef
  • Google Scholar
  • View reference in article

• 69.

  KällénGPauliW. On the mathematical structure of renormalizable field theories. Mat Fys Medd Dan Vidensk Selsk (1951) 24.

  • Google Scholar
  • View reference in article

• 70.

  YennieDFrautschiSSuuraH. The infrared-divergence phenomena and high-energy processes. Ann Phys (1961) 13:379–452. 10.1016/0003-4916(61)90151-8

  • CrossRef
  • Google Scholar
  • View reference in article

• 71.

  KasprzakJRichardMKundermannSBaasAJeambrunPKeelingJMJet alBose–einstein condensation of exciton polaritons. Nature (2006) 443:409–14. 10.1038/nature05131

  • Pubmed Abstract
  • CrossRef
  • Google Scholar
  • View reference in article

• 72.

  ImadaMFujimoriATokuraY. Metal-insulator transitions. Rev Mod Phys (1998) 70:1039–263. 10.1103/RevModPhys.70.1039

  • CrossRef
  • Google Scholar
  • View reference in article

• 73.

  AokiK. Introduction to the non-perturbative renormalization group and its recent applications. Int J Mod Phys B (2000) 14:1249–326. 10.1142/s0217979200000923

  • CrossRef
  • Google Scholar
  • View reference in article

• 74.

  StreetR. Hydrogenated amorphous silicon. Cambridge, UK: Cambridge University Press (1991). 10.1017/CBO9780511623086

  • CrossRef
  • Google Scholar
  • View reference in article