Fermion-Boson Duality and Quantum Field Theory Using an Extended Dirac Equation

Quantum Field Theory Using Fermion-Boson Duality and Extended Dirac Equation Hirokazu Maruyama Independent Researcher Kobe, Hyogo, Japan [email protected] January 15, 2025 Abstract This paper proposes a theory of fermion-boson duality in which field statistics change continuously according to energy scales. The theory demonstrates that gauge fixing and ghost fields become unnecessary through the implementation of fermion-boson duality, which naturally eliminates redundant degrees of freedom. Furthermore, this study introduces 256-dimensional gamma matrices to incorporate gravity into the quantum field theoretical framework. A phase transition mechanism is proposed where electrons exhibit bosonic properties inside atoms while maintaining fermionic characteristics outside, with photons showing the opposite behavior. This framework naturally avoids ultraviolet divergences without artificial regularization. The theory is consistent with existing experimental data and predicts new phenomena in high-energy regions. Applications to muon anomalous magnetic moment and lattice simulations are also discussed, suggesting a unified understanding of gauge theories and gravity through statistical transmutation. Keywords: Phase transition, Statistical inversion, Quantum field theory, Gauge theory, Quantum gravity, Duality principle, Extended Dirac equation, Gamma matrices 1 1 Introduction Quantum field theory is one of the most important achievements in 20thcentury physics and serves as the fundamental theory of particle physics, providing the most successful theoretical framework for describing nature’s basic interactions. In particular, quantum electrodynamics (QED) agrees with experiments with remarkable precision[1, 2], and quantum chromodynamics (QCD) has successfully explained the essential properties of strong interactions[3, 4, 5]. The problem of statistics in quantum field theory remains an active research topic[6, 7]. In particular, the behavior of statistics in the high-energy limit has attracted attention in relation to quantum gravity[8]. Additionally, observations of similar phenomena in condensed matter physics[9] suggest the possibility of verifying this theory’s predictions. However, standard quantum field theory retains the following fundamental challenges: 1.1 Current State and Challenges of Quantum Field Theory First, the gauge fixing and ghost field introduction required for gauge field quantization obscures the physical interpretation of the theory. For example, in quantum chromodynamics (QCD), gauge fixing conditions must be imposed on each of the eight gluon fields, and corresponding ghost fields must be introduced. This significantly complicates the theory’s structure and makes physical interpretation difficult. Second, a unified description of fields with different statistics - fermions and bosons - has not been achieved. While supersymmetry theory provides one answer to this problem, it has yet to be experimentally verified. Moreover, supersymmetry theory requires partner particles corresponding to each particle, which increases the theory’s complexity. Third, consistent coupling with gravitational fields is difficult. Quantization of gravity based on general relativity faces the problem of ultraviolet divergences, which cannot be resolved within the framework of conventional field theory. This issue poses serious challenges to the theory’s consistency in the high-energy limit. Behind these problems lie the following fundamental questions: 1. Can field statistics change with energy scale? - At low energies, electrons behave as fermions and follow the Pauli exclusion principle - However, at ultra-high energies, this picture may change - In superconductivity phenomena, electrons show effectively bosonic behavior through Cooper pair 2 formation - The possibility of such statistical changes occurring at a more fundamental level needs to be examined 2. What is the relationship between gauge principle and gravity? - Gauge fields and gravitational fields appear to be based on different symmetries - However, they may be fundamentally connected - A unified description at high energies is needed 3. Do ultraviolet divergences have physical meaning? - Current theory requires artificial regularization - This may suggest incompleteness of the theory - May be naturally resolved in a more fundamental theory 1.2 Approach of This Research In this research, we propose fermion-boson duality as a new approach to these challenges. This duality is mathematically formulated using an extended Dirac equation and has the following characteristics: 1. Demonstrates the possibility that field statistics can change dynamically depending on energy scale 2. Enables formulation without requiring gauge fixing or ghost fields 3. Provides a mechanism where gravitational fields naturally emerge as quantum effects Specifically, we introduce 256 ×256 extended gamma matrices and use them to generalize the conventional four-dimensional Dirac equation. This extension provides the following advantages: 1. Unified description of fermionic and bosonic degrees of freedom 2. Natural realization of physical degrees of freedom (two transverse components) of gauge fields 3. Automatic avoidance of ultraviolet divergences 1.3 Structure of the Paper This paper is structured as follows: First, we provide a detailed overview of the basic structure and challenges of conventional quantum field theory. In particular, we discuss the necessity of gauge fixing and ghost field introduction, the problem of ultraviolet divergences, and difficulties in coupling with gravitational fields. Next, we introduce bosonic and fermionic gamma matrices and duality theory, developing their basic mathematical structure. We construct 256 × 256 extended gamma matrices and formulate the extended Dirac equation based on them. 3 We then discuss the quantum theoretical description and symmetry of the metric tensor, demonstrating the possibility of natural coupling with gravitational fields. Following this, we examine specific applications of this theory: •Construction of extended quantum electrodynamics •Construction of extended quantum chromodynamics •Fundamental theory of statistical inversion in quantum chromodynamics (QCD) •Mechanism of gravitational field emergence 2 Basic Structure of Conventional Quantum Field Theory Quantum field theory is constructed based on the fundamental idea of field quantization. In this section, we first overview the basic structure of conventional theory and then discuss in detail the challenges currently faced. 2.1 Basic Structure of Quantum Electrodynamics (QED) Quantum electrodynamics is the most successful quantum field theory describing electromagnetic interactions. Its basic Lagrangian is given by: LQED =ψ(iγµDµ−m)ψ−1 4FµνFµν (1) This Lagrangian consists of the following three fundamental terms: 1. The first term is the kinetic term of the Dirac field, describing the free motion of charged particles and their interaction with the electromagnetic field. Here, Dµ=∂µ+ieAµrepresents the covariant derivative, realizing minimal coupling with the electromagnetic field. 2. The second term is the mass term, characterizing the fermion mass. 3. The third term is the kinetic term of the electromagnetic field, containing the field strength tensor given by Fµν =∂µAν−∂νAµ. This theory is invariant under the following gauge transformations: ψ→e−ieα(x)ψ(2a) Aµ→Aµ+∂µα(x) (2b) 4 This gauge invariance is a fundamental symmetry of the theory and leads to charge conservation. However, this structure includes two important technical challenges: 1. Infrared divergences in the photon propagator 2. Ultraviolet divergences in perturbation theory To address these problems, conventional theory has required the following technical prescriptions: 2.1.1 Gauge Fixing Term The gauge fixing term added to the QED Lagrangian takes the form: LGF =−1 2ξ(∂µAµ)2(3) Here, ξis the gauge parameter, and in the usual Lorenz gauge, we choose ξ= 1. This term allows proper definition of the photon propagator and enables path integral calculations. 2.1.2 Renormalization Prescription To handle terms containing divergences appropriately, we introduce the following renormalization transformation: Lren =Z2ψ(iγµDµ−m)ψ−Z3 4FµνFµν (4) Here, Z2represents the fermion field renormalization factor, and Z3represents the photon field renormalization factor. By appropriately choosing these factors, the theory’s finiteness is guaranteed. This is the standard prescription of quantum electrodynamics and yields results that agree with experiments with remarkable precision. 2.2 Basic Structure of Quantum Chromodynamics (QCD) Quantum chromodynamics is a non-abelian gauge theory describing strong interactions. Its basic Lagrangian is: LQCD = nf X f=1 qf(iγµDµ−mf)qf−1 4Ga µνGaµν (5) Here, the covariant derivative takes the form: Dµ=∂µ+igsTaGa µ(6) 5 The characteristic structure of this theory appears in the following nonabelian field strength tensor[10]: Ga µν =∂µGa ν−∂νGa µ+gsfabcGb µGc ν(7) Due to QCD’s non-abelian nature, the following technical challenges arise: 1. Necessity of gauge fixing: to eliminate redundant degrees of freedom in the gluon field 2. Introduction of Faddeev-Popov ghost fields[11]: to cancel unphysical degrees of freedom 3. Realization of asymptotic freedom: to explain the decrease of coupling constant at high energies[3, 4] To address these problems, the following terms needed to be introduced in conventional theory: 2.2.1 Gauge Fixing Term To control the longitudinal components of the gluon field Ga µ, LGF =−1 2ξ(∂µGaµ)2(8) is introduced. Here, ξis the gauge parameter, and in the Lorenz gauge, we choose ξ= 1. 2.2.2 Faddeev-Popov Ghost Term To ensure the unitarity of non-abelian gauge theory, Lghost =ca∂µDµca, Dµca=∂µca+gfabcGbµcc(9) is introduced. Here, ca, caare Faddeev-Popov ghost fields, fabc are structure constants, and gis the coupling constant. 2.2.3 BRST Symmetry The entire theory including the above gauge fixing and ghost terms is invariant under the following BRST transformations[12]:[13] δBGa µ=Dµca(10a) δBca=−g 2fabccbcc(10b) δBca=1 ξ∂µGa µ(10c) This symmetry ensures the physical unitarity of the theory. 6 2.3 Challenges and Limitations of Conventional Theory Conventional quantum field theory contains the following fundamental challenges: 1. Problems with Gravitational Field Consistency - Perturbative nonrenormalizability - Background dependence - Energy scale hierarchy 2. Technical Complexity of Mathematical Structure - Arbitrariness in gauge fixing - Introduction of ghost degrees of freedom - Complexity in handling divergences 3. Opacity of Physical Interpretation - Physical meaning of virtual processes - Existence of unphysical degrees of freedom - Complexity of vacuum structure These challenges suggest the necessity of a more fundamental theory. 3 Bosonic and Fermionic Gamma Matrices and Duality Theory In this section, we propose a theoretical framework that can naturally express the duality between electrons and photons by introducing two types of gamma matrices with different statistics. 3.1 Introduction of Two Types of Gamma Matrices In this research, in addition to the conventional Dirac gamma matrices γµ, we introduce another representation γ′ µand define the following two operators: γµ(Fermionic type) (11a) ωµ=γµ+γ′ µ 2(Bosonic type) (11b) These operators satisfy the following anticommutation relations: {ω1, ω1}={ω2, ω2}=−2I4(12a) {ω0, ω0}={ω3, ω3}= 0 (12b) {ωi, ωj}= 0 (i=j) (12c) Here, i, j ∈ {0,1,2,3}represent spinor indices, and I4is the 4×4 identity matrix. This algebraic structure gives rise to ”half-Hermitian components” and ”half-anti-Hermitian components” different from conventional γµ, resulting in operators with different properties in time and space directions. 7 3.2 Fundamental Properties The following properties are important in this theory: 1. Vanishing of traces: tr(ωµ) = 0 (for all µ) (13) 2. Hermiticity: ω† µ=(ωµ, µ = 0,3 −ωµ, µ = 1,2(14) This results in fundamentally different symmetries and invariances between ω0, ω3and ω1, ω2, which connect to physical mechanisms such as statistical changes and ultraviolet cutoffs that will be shown later. 3.3 Explicit Representation of Bosonic Gamma Matrices Based on the attached calculation results, the explicit matrix representation of the bosonic gamma matrices is given by: 8 ω0=    1 20i 20 01 20−i 2 i 20−1 20 0−i 20−1 2    (15a) ω1=i    0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0    (15b) ω2=    0001 0 0 −1 0 0100 −1 0 0 0    (15c) ω3=    1 20i 20 01 20−i 2 i 20−1 20 0−i 20−1 2    (15d) As confirmed by numerical calculations using Mathematica, these matrices strictly satisfy the above anticommutation relations. 3.4 Mathematical Structure of Duality In quantum state space, electrons and photons each possess two possible statistical natures: |ψtotal⟩=|ψeF⟩+|ψeB⟩+|ψγF⟩+|ψγB⟩(16) Here, -|ψeF⟩: Fermionic electron state - |ψeB⟩: Bosonic electron state - |ψγF⟩: Fermionic photon state - |ψγB⟩: Bosonic photon state The realization of these states is described by the transition function T(E). The transition function is defined as: T(E) = 1 −exp−E2/E2 c(17) 9 •Manifests inside atoms and high-pressure regions, providing new physical degrees of freedom 5.4 (3) Photon Field Kinetic Term: −1 4Fµν Fµν •Describes photon motion in normal states, representing properties of transverse two-component bosonic photons •Becomes dominant in low-energy regions, ensuring consistency with classical electromagnetism •Describes fundamental properties of photons while preserving gauge symmetry 5.5 (4) Fermionic Photon Field Kinetic Term: −1 4Tµν Tµν •Inside atoms, the ordinary photon field kinetic term FµνFµν has duality with fermionic TµνTµν •This energy-momentum tensor TµνTµν generates the metric tensor gµν •Through this, local gravitational fields naturally emerge inside atoms This enables a unified description of quantum electrodynamics and gravity. In particular, inside atoms: 1. Realization of duality between photon field kinetic terms FµνFµν and TµνTµν 2. Natural emergence of metric tensor through energy-momentum tensor 3. Generation of gravitational field through spacetime curvature is achieved. 5.6 Concrete Calculation Example: Compton Scattering Based on the attached calculation results, we present Compton scattering as a concrete example. This calculation provides an important example for quantitatively understanding the effects predicted by the theory. 16 5.6.1 Calculation in Minkowski Spacetime First, in Minkowski metric gµν = diag(−1,1,1,1), the scattering amplitude is: M=u(p′)"γµ/ p+/ k+m (p+k)2−m2γν+γν/ p−/ k′+m (p−k′)2−m2γµ# ×u(p)ϵµ(k)ϵ∗ ν(k′) (36) Numerical calculations using Mathematica have confirmed that this amplitude completely agrees with the conventional Klein-Nishina formula:[14] dσ dΩ=α2 2m2ω′ ω2ω′ ω+ω ω′−sin2θ(37) 5.6.2 Calculation in Curved Spacetime For general metrics, the scattering amplitude receives modifications. According to calculations: dσ dΩ=dσKN dΩ[1 + κT(E)FµνFµν] (38) Here, the second term represents spacetime modification effects through TµνTµν arising from the duality of FµνFµν possessed by electrons and photons. In particular, while this effect theoretically exists in the region E∼Ec, its influence is extremely small, and direct observation with current experimental technology is predicted to be difficult. 5.7 Proof of Gauge Fixing Unnecessity The unnecessity of gauge fixing in this theory is proved in three steps. 5.7.1 Step 1: Automatic Restriction of Degrees of Freedom In the expansion using bosonic gamma matrices Ωµ, the photon field is: ΩµAµ(x) = ΩµZd3p (2π)32p0[aµ(p)e−ipx +a† µ(p)eipx] (39) Calculating the square of this expression: (ΩµAµ)2= (A1)2+ (A2)2(40) This demonstrates that only two transverse components automatically remain. 17 5.7.2 Step 2: Preservation of Gauge Invariance The theory’s gauge invariance: δAµ=∂µα, δψ =ieαψ (41) Under this transformation, the action is invariant: δS = 0 (42) This is realized without introducing gauge fixing. 5.7.3 Step 3: Proof of Unitarity The unitarity of the scattering matrix: SS†=S†S= 1 (43) This is satisfied without introducing gauge fixing or ghost fields. Details of the proof are provided in the appendix. 5.8 Concrete Predictions of the Theory This theory provides the following experimentally verifiable predictions: 1. Modification of scattering cross-section: σ(E) = σQED(E)[1 + κT(E)] (44) 2. Contribution to anomalous magnetic moment: ae=α 2π[1 −T(me)] + O(α2) (45) 3. Vacuum polarization effects: Πµν(q) = (q2gµν −qµqν)Π(q2)[1 −T(q)] (46) These effects have magnitudes that can be verified by current precision measurement experiments. 5.9 Construction of Extended Quantum Chromodynamics The extension of fermion-boson duality to quantum chromodynamics (QCD) possesses a richer physical structure. This is due to the theory’s non-abelian nature and strong coupling. 18 5.10 Extended Lagrangian The extended QCD Lagrangian takes the form: LextQCD = nq X f=1 qfhi gµνΓνDµ+i gµνΩµDν−mfiqf −1 4Ga µν Ga µν −1 4e Ga µν e Ga µν (47) In this expression: 1. gµν is the metric tensor 2. Γµare fermionic type gamma matrices 3. Ωµare bosonic type gamma matrices 4. e Ga µν is the fermionic type gluon field strength tensor 5. Duality is introduced for both quarks and gluons In the extended QCD Lagrangian: −1 4e Ga µν e Ga µν (48) represents the kinetic term of the fermionic gluon field, which possesses duality with the ordinary gluon field kinetic term Ga µνGa µν. Through this duality, like in QED, it affects the spacetime metric through the energy-momentum tensor, giving rise to gravitational effects. This structure suggests that the statistical inversion of gluon fields at high energies causes spacetime curvature and is observed as a gravitational field. 5.11 Physical Degrees of Freedom of Gluons In conventional QCD, the gluon field Ga µhad 4 ×8=32 degrees of freedom, and gauge fixing and ghost fields were needed to restrict these to 16 physical degrees of freedom (2 polarizations ×8 colors). In this theory: 1. Degrees of freedom are automatically restricted by bosonic gamma matrices 2. Non-abelian nature is preserved while only physical degrees of freedom remain 3. Possibility of new physical effects in the strong coupling region 19 6 Fundamental Theory of Statistical Inversion in Quantum Chromodynamics (QCD) 6.1 Theoretical Framework The effective action describing statistical inversion in QCD takes the form: L=ψ(igµνΓνDµ+igµνΩµDν−m)ψ−1 4GµνGµν −1 4TµνTµν (49) Here, the transition function T(E) is defined as: T(E) = 1 −exp−E2/E2 c(50) and the following is naturally realized according to energy scale: 1. Low-energy region (E≪Ec): - Fermionic quarks - Bosonic gluons - T(E)≈0 2. High-energy region (E≫Ec): - Bosonic quarks - Fermionic gluons - T(E)≈1 6.2 Transition Function and State Vectors Corresponding to this transition function, quark and gluon states are: |ψq(E)⟩= [1 −T(E)]|ψF⟩+T(E)|ψB⟩(51a) |ψg(E)⟩= [1 −T(E)]|ψB⟩+T(E)|ψF⟩(51b) The coupling constant is: αs,eff(E) = αs(E)[1 −T(E)] (52) This form naturally realizes asymptotic freedom. 6.3 New Structure of Non-Abelian Gauge Theory The non-abelian nature of QCD is preserved in the following modified form: [Dµ, Dν] = igsGa µνTa[1 −T(E)] + igse Ga µνTaT(E) (53) Important points here are: 1. Preservation of Non-Abelian Nature - Maintenance of group structure - Conservation of color symmetry - Realization of gauge invariance 2. Statistics-Dependent Coupling - Change with energy scale - Possibility of new bound states - Prediction of phase transition phenomena 20 6.4 Concrete Calculation Examples 6.4.1 Gluon Propagator The modified gluon propagator is: Dab µν(k) = −iδab k2gµν −kµkν k2[1 −T(k)] (54) Characteristics of this form: 1. No need for gauge fixing 2. Propagation of transverse modes only 3. Natural suppression at high energies 6.4.2 Quark Self-Energy The quark self-energy term is: Σ(p) = −ig2 sCFZd4k (2π)4γµ/ k+/ p (k+p)2γνDµν(k)[1 −T(k)] (55) This gives a finite result: Σ(p) = αsCF 4π/ pln E2 c −p2(56) 6.5 New Understanding of Asymptotic Freedom Asymptotic freedom in this theory is understood as a combination of three mechanisms: 1. Reorganization of degrees of freedom through statistical inversion βQCD =−α2 s 4π(11Nc−2Nf)[1 −T(E)] (57) 2. Bosonization at high energy lim E→∞ T(E) = 1 (58) 3. Natural decrease of coupling constant lim E→∞ αs,eff(E) = 0 (59) 21 6.6 Theory’s Extensibility While providing important implications for unsolved problems in particle physics, this theoretical framework also holds potential for further extension. Particularly noteworthy directions include unification with electroweak theory and transcendence of the supersymmetry paradigm. In unification with electroweak theory, particle statistical changes may provide a new mechanism for spontaneous symmetry breaking. This approach suggests a new framework alternative to the conventional Higgs mechanism and is expected to provide a natural generation mechanism for the electroweak scale. Moreover, an important point is that this theory presents a fundamental alternative to supersymmetry theory. Instead of requiring the introduction of supersymmetric partner particles as demanded by conventional supersymmetry theory, it provides a more natural mechanism of statistical self-change through duality possessed by particles themselves. This change occurs dynamically according to energy scale and brings about physically observable effects. The most notable feature of this theory is that it can explain physical effects through changes in the properties of existing particles themselves without introducing new particles. This feature brings two important aspects: theoretical economy and experimental verifiability. From the perspective of theoretical economy, it is possible to avoid introducing extra particles and minimize the number of necessary parameters. This enables a more natural unified description. In particular, it can show a path toward unified understanding of physical laws in a form that follows Occam’s razor principle. From the perspective of experimental verifiability, it is important that phenomena can be observed as changes in the properties of existing particles. This enables clear physical predictions and more direct experimental approaches. In particular, it is expected that theory verification will be possible through detailed observation of existing particle behavior in current accelerator experiments and precision measurements. Thus, while remaining faithful to fundamental principles of physics, this theory provides a new perspective that transcends conventional theoretical frameworks. This not only enables new approaches to unsolved problems in particle physics but also opens a path toward deeper understanding of physical laws. 22 7 Mechanism of Gravitational Field Generation In this section, we discuss the mechanism by which gravitational fields naturally emerge through the duality of gauge field kinetic terms. 7.1 Gravity Generation in QED In the electromagnetic field Lagrangian, the ordinary photon field kinetic term FµνFµν possesses duality with TµνTµν in high-density regions inside atoms. This energy-momentum tensor term affects the metric tensor gµν and brings about spacetime curvature. Particularly inside atoms, as the photon field kinetic term changes from FµνFµν to TµνTµν, local gravitational fields emerge through accompanying modifications of gµν. This process demonstrates the mechanism by which gravitational fields naturally emerge from gauge field duality. 7.2 Gravity Generation in QCD A similar mechanism is realized in QCD. The gluon field kinetic term Ga µνGaµν shows duality with e Ga µν e Gaµν in high-density regions. This duality, like in the QED case, generates gravitational effects through the metric tensor. Particularly inside nucleons, as the gluon field kinetic term changes from Ga µνGaµν to e Ga µν e Gaµν, stronger gravitational fields emerge. This suggests enhancement of gravitational effects at the nucleon scale. 7.3 Physical Significance and Implications This mechanism provides several important insights into the fundamental nature of gravity. First, it provides a natural explanation for the hierarchy problem regarding the strength of gravity. At the individual particle level, the effect through duality is extremely small, explaining why gravity is remarkably weaker than the other three forces. However, when many particles gather, this effect accumulates and becomes observable as macroscopic gravity. Second, it provides a new understanding of the origin of gravity. In this theory, gravity emerges from the gauge field duality inherently possessed by electrons and quarks. This provides a natural explanation for the existence of gravity as a fundamental property of matter and enables intuitive understanding of the relationship between particle properties and gravity. 23 Furthermore, this theory contains important cosmological implications. Gravitons arising through duality may possess mass, similar to photons in superconductors. This feature provides a new perspective on galaxy rotation curve anomalies and the dark matter problem, suggesting fundamental modifications to the long-range behavior of gravity. 7.4 Observability Regarding the observability of gravitational effects predicted by this theory, the following characteristics are important. At the atomic scale, the effects are minute and direct observation is difficult. On the other hand, stronger effects are predicted inside nucleons, and they may be observable as collective effects in high-density matter. Furthermore, at cosmological scales, they may appear as effects of massive gravitons. These effects provide a new perspective on the unified understanding of particle interactions and gravity, while suggesting new approaches to unsolved problems in astrophysics. In particular, they are expected to bring essential progress to understanding the quantum nature of gravity and its behavior at cosmological scales. 8 Conclusion In this research, we have proposed fermion-boson duality and extended Dirac equations as one possibility in quantum field theory. We attempted a formulation using 256 ×256 gamma matrices, showing one approach toward unified description of fermion fields and boson fields that have traditionally been treated separately. From this theoretical attempt, the following implications were obtained: First, the possibility of mathematical description of statistics. Using the extended Dirac equation with bosonic gamma matrices Ωµand fermionic gamma matrices Γν, we attempted to express dynamic changes in statistics dependent on energy scale. In particular, we were able to present one theoretical possibility regarding the reorganization of physical degrees of freedom in the high-energy limit. Second, the proposal of a new perspective on gauge theory. We demonstrated the possibility of formulation not requiring the gauge fixing and ghost fields that have been introduced in conventional theory. In this framework, it was suggested that physical degrees of freedom of photons and gluons might be naturally restricted to two transverse components, providing one 24 direction toward simplification of the physical picture. Additionally, a new interpretation possibility was suggested for QCD’s asymptotic freedom. Third, we were able to present one perspective on the relationship with gravitational fields. In this theoretical framework, gravitational fields are suggested to emerge naturally from gauge field duality. This may provide one perspective toward quantum theoretical understanding of gravity. 9 Acknowledgments In conducting this research, email discussions with university professors and associate professors specializing in particle physics had a decisive influence on the conception of fermion-boson duality that forms the core of this paper. I deeply thank both professors for the many insights into the mathematical structure and physical meaning of the theory obtained through deep discussions with them. I would also like to express my sincere gratitude to Claude, a generative AI, for providing editorial support. References [1] Schwinger, J., “On Quantum-Electrodynamics and the Magnetic Moment of the Electron,” Phys. Rev. 73, 416–417 (1948). [2] Kinoshita, T., “Higher-Order QED Calculations for the Electron and Muon Anomalous Magnetic Moments,” Rev. Mod. Phys. 81, 865–934 (2009). [3] Gross, D. J., “Ultraviolet Behavior in Non-Abelian Gauge Theories,” Phys. Rev. Lett. 30, 1343–1346 (1973). [4] Politzer, H., “Reliable Perturbative Results for Strong Interactions?,” Phys. Rev. Lett. 31, 581–583 (1973). [5] Wilczek, F., “Asymptotic Freedom in Parton Language,” Phys. Rev. Lett. 31, 621–623 (1973). [6] Witten, E., “Topological Quantum Field Theory,” Commun. Math. Phys. 174, 155–186 (1995). [7] Gaiotto, D., “N=2 Dualities,” Adv. Theor. Math. Phys. 19, 1163–1205 (2015). [arXiv:1412.3478] 25 D.1 Theoretical Framework D.1.1 Conventional QED Path Integration For the standard QED Lagrangian LQED =ψiγµDµ−mψ−1 4FµνFµν (80) when treating in path integration, the photon field Aµhas redundancy under gauge transformations Aµ→Aµ+∂µΛ. Therefore, to rigorously define: Z=ZDAµDψDψexp{iS}(81) it was necessary to introduce gauge fixing terms and Faddeev-Popov ghosts. D.1.2 New Formulation In this theory, by using bosonic gamma matrices Ωµ, we restrict the photon field to physical two components from the beginning: ZextQED =ZDψDψD(Ω-type) expiSextQED[ψ, ψ, Ω](82) The advantages of this form are: 1. Physical degrees of freedom (two transverse components) are automatically selected 2. No need to introduce gauge fixing 3. Ghost fields unnecessary D.2 Application to Loop Calculations D.2.1 Photon Propagator When calculating one-loop corrections in conventional QED, the internal line photon propagator took the ξ-dependent form: −i k2+iεhgµν −(1 −ξ)kµkν k2i(83) In contrast, in this theory, the photon propagator takes the simple form: DΩ(k, λ) = −i k2+iε δλλ′(84) This results in: 1. Automatic elimination of longitudinal and time components 2. No need for gauge fixing parameter 3. Essential simplification of calculations 32 D.3 Automatic Regularization of Divergences In the new formulation, loop integrals are naturally regularized as follows: Zd4k (2π)4 1 k2→Zd4k (2π)4 1−T(k/Ec) k2(85) This modification has the following characteristics: 1. Natural suppression in high momentum region 2. No need for cutoff parameter introduction 3. Preservation of gauge invariance D.4 Calculation Example of Vacuum Polarization As a concrete example, we show the calculation of the vacuum polarization tensor: Πµν(q) = ie2Zd4k (2π)4Tr γµ / k k2γν / k+/ q (k+q)2[1 −T(k)] (86) The result: Πµν(q) = (q2gµν −qµqν)α 3πln E2 c −q2(87) This result demonstrates: 1. Finite expression containing no divergences 2. Preservation of gauge invariant structure 3. Reproduction of standard QED in low energy limit E Detailed Calculation of Muon Anomalous Magnetic Moment E.1 Derivation of Difference from Standard Theory The difference between experimental value and standard theory prediction for muon anomalous magnetic moment: ∆aµ=aexp µ−aSM µ= (251 ±59) ×10−11 (88) To explain this difference, we consider the effect of statistical inversion. 33 E.2 Calculation of Statistical Inversion Correction Modified anomalous magnetic moment: aµ=aQED µ[1 −T(mµ)] + anew µT(mµ) (89) Here, T(mµ) is the transition function: T(mµ) = 1 −exp−m2 µ/E2 c(90) The calculation proceeds in the following steps: 1. Modification of photon propagator: Dµν(k) = −gµν k2[1 −T(k)] (91) 2. Calculation of vertex function: Γµ(p′, p) = γµF1(q2) + iσµνqν 2mF2(q2) (92) 3. Derivation of form factors: F1(q2) = 1 + α 2πZ1 0 dx 2x2(1 −x) x2+ (1 −x)(q2/m2)[1 −T(m√x)] (93a) F2(q2) = α 2πZ1 0 dx x2(2 −x) x2+ (1 −x)(q2/m2)[1 −T(m√x)] (93b) E.3 Evaluation of Higher-Order Corrections Second and higher-order correction terms take the following forms: 1. Two-photon exchange: a(2γ) µ=α π2Z1 0 dx Z1 0 dy f(x, y)[1 −T(mµ√xy)] (94) 2. Hadronic vacuum polarization: ahad µ=α π2Z∞ 4m2 π ds K(s)σ(s)[1 −T(√s)] (95) 3. Weak interaction correction: aweak µ=GFm2 µ 8√2π25 3+α 4π[1 −T(MW)] (96) 34 E.4 Application to Tau Particles Applying the same calculation to tau particles: ∆aτ ∆aµ =T(mτ) T(mµ)≈1−e−m2 τ/E2 c 1−e−m2 µ/E2 c(97) This ratio gives an experimentally verifiable prediction: ∆aτ∼∆aµ×mτ mµ2 (98) These calculations demonstrate that the statistical inversion theory gives predictions consistent with experimental results. F Comparison and Proof of Regularization Methods in QCD F.1 Detailed Comparison of Various Regularization Methods F.1.1 Dimensional Regularization In conventional dimensional regularization, spacetime dimension is analytically continued to d= 4 −2ϵ: Zd4k (2π)4→µ2ϵZddk (2π)d(99) Specific divergent integrals reduce to the form: I=µ2ϵ 16π22 ϵ+ ln(4π)−γE+O(ϵ)(100) Problems with this method: •Difficulty in handling chiral symmetry •Introduction of unphysical parameter ϵ •Necessity of limit operations 35 F.1.2 Pauli-Villars Regularization Introduction of regulator fields with higher derivatives: [15] LPV =ψ(i/ ∂−m)ψ+ N X i=1 ciΨi(i/ ∂−Mi)Ψi(101) Here, coefficients ciand masses Misatisfy: N X i=1 ci= 1 (102a) N X i=1 ciM2n i= 0 (n= 0,1, . . . , N −1) (102b) F.1.3 Lattice Regularization Discretization of spacetime with lattice spacing a: Zd4x→a4X x (103a) ∂µf(x)→1 a[f(x+aˆµ)−f(x)] (103b) Link variables: Uµ(x) = exp (iagAµ(x)) (104) F.2 Proof of Automatic Regularization Through Statistical Inversion In this theory, introduction of the transition function T(E) realizes the following natural regularization: Zd4k (2π)4 1 k2→Zd4k (2π)4 1−T(k) k2(105) We prove the properties of this regularization in the following steps. F.2.1 Step 1: Convergence at High Momentum For k≫Ec: lim k→∞[1 −T(k)] = 0 (106) This ensures ultraviolet convergence of the integral. 36 F.2.2 Step 2: Preservation of Gauge Invariance Under gauge transformations: Aµ→Aµ+∂µα(107a) ψ→e−igαψ(107b) we demonstrate that the action remains invariant. After transformation: Zd4k (2π)4 1−T(k) k2eik·x=Zd4k (2π)4 1−T(k) k2eik·x(108) F.2.3 Step 3: Proof of Ward Identity Current conservation: ∂µjµ= 0 (109) We show this holds including quantum corrections: kµΠµν(k) = 0 (110) F.3 Details of Divergence Avoidance Mechanism F.3.1 Structure of General Divergent Integrals General divergent integral at nloops: In=Zn Y i=1 d4ki (2π)4 P(ki) Q(ki, pj)(111) F.3.2 Modification by Statistical Inversion Modification by transition function: Imod n=Zn Y i=1 d4ki (2π)4 P(ki) Q(ki, pj) n Y i=1 [1 −T(ki)] (112) This integral has the following properties: 1. Natural convergence in ultraviolet region 2. Preservation of symmetries 3. Maintenance of unitarity 37 G Comparison of Statistical Inversion and Coupling Constant Behavior in QCD and QED G.1 Differences in Gauge Group Structure Quantum chromodynamics and quantum electrodynamics have fundamental differences in their underlying gauge group structures. These differences determine the nature of interactions and high-energy behavior in both theories. G.1.1 QCD: Characteristics of Non-Abelian Gauge Theory Quantum chromodynamics is a non-abelian gauge theory with SU(3) as its fundamental symmetry. The theory’s characteristics are summarized in the following four points: 1. Adoption of SU(3) as gauge group naturally introduces three color degrees of freedom 2. Existence of non-vanishing structure constants fabc = 0 ensures theory’s non-abelian nature 3. As a direct consequence of this non-abelian nature, gluon field selfinteractions are inevitably introduced 4. Rich physical structure of strong interactions is realized through gluons themselves carrying color charge G.1.2 QED: Characteristics of Abelian Gauge Theory In contrast, quantum electrodynamics is an abelian gauge theory based on U(1) group. The theory’s characteristics form a striking contrast with QCD: 1. Simple structure of gauge group U(1) ensures existence of single charge 2. Absence of structure constants makes theory inherently abelian 3. As a direct consequence of this abelian nature, photon field selfinteractions do not exist 4. Photons function only as electromagnetic interaction mediators as they carry no charge G.2 Behavior in High-Energy Limit In quantum field theory including statistical inversion, QCD and QED show fundamentally different behaviors in the high-energy limit. 38 G.2.1 QCD Case The high-energy limit behavior in quantum chromodynamics is characterized by the combination of three physical mechanisms: 1. As energy approaches infinity (E→ ∞), effective coupling constant vanishes through transition function approaching 1: αs,eff(E)→0 (113) 2. Anti-screening effect through gluon self-interaction plays essential role 3. Statistical inversion acts as mechanism promoting this coupling decrease G.2.2 QED Case In contrast, quantum electrodynamics shows completely different behavior in high-energy limit: 1. As energy approaches infinity, effective coupling constant αeff(E) increases 2. Screening effect through vacuum polarization becomes dominant 3. Statistical inversion acts as mechanism promoting this coupling increase These characteristics demonstrate how the same statistical inversion mechanism can lead to qualitatively different results depending on the underlying gauge structure. H Relationship with Anyonic Phase Factors The wavefunction of anyons acquires the following phase factor under exchange of two particles: ψ(r1, r2)→eiθψ(r2, r1) (114) Here the phase angle θtakes values in the range: 0≤θ≤π(115) where θ= 0 corresponds to bosons and θ=πcorresponds to fermions. On the other hand, the transition function T(E) in this theory: T(E) = 1 −exp−E2/E2 c(116) describes a continuous change dependent on energy. 39 These have the following correspondence relations: 1. Similarity with anyonic phase factor: eiθ ↔[1 −T(E)] + iT(E) (117) 2. Correspondence of limits: θ= 0 (boson) ↔T(E) = 0 (low energy) (118a) θ=π(fermion) ↔T(E) = 1 (high energy) (118b) 3. Superposition representation of states: |ψ(E)⟩= [1 −T(E)]|ψF⟩+T(E)|ψB⟩(119a) ∼cos(θ/2)|ψF⟩+isin(θ/2)|ψB⟩(119b) Thus, the transition function T(E) can be interpreted as a natural generalization of the anyonic phase factor. However, important differences include: 1. While anyonic phase factors are limited to two-dimensional systems, T(E) is defined even in four-dimensional spacetime 2. While anyonic phase factors arise from geometric properties of phase space, T(E) represents dynamic effects dependent on energy scale 3. While anyonic phase factors relate to exchange statistics, T(E) describes changes in the statistical nature of the fields themselves I About Obtaining Mathematica Calculation Code I.1 Code Structure The complete codebase is organized as follows: •QED Processes –Compton scattering –Bhabha scattering –Møller scattering –Muon pair production •Theoretical Foundations –Properties of ωgamma matrices 40 –Definition and verification of 4-dimensional and 256-dimensional gamma matrices •Detailed Scattering Process Calculations –Setting up kinematic variables –Calculation of scattering amplitudes –Derivation of cross sections –Comparison with Klein-Nishina formula –High-energy limit behavior •Other Interactions –Muon decay process These calculation codes demonstrate concrete applications of this theory while enabling comparative verification with conventional quantum field theory. In particular, the construction of bosonic gamma matrices and calculation of Compton scattering play important roles in confirming basic predictions of the theory. J Data Availability The supplementary data and Mathematica calculations associated with this paper are publicly available at: •Figshare Repository (with DOI): https://doi.org/10.6084/m9.figshare.28208264 •GitHub Repository: https://github.com/HM-Physics/FermionBosonDuality_QFT 41