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

Fermion-Boson Duality and Quantum Field Theory Using an Extended Dirac Equation Hirokazu Maruyama Independent Researcher Kobe, Hyogo, Japan [email protected] November 14, 2025 Abstract We propose a framework of fermion–boson duality in which the statistical character of quantum fields varies continuously with the energy scale. Within this framework, the implementation of the duality renders gauge fixing and Faddeev–Popov ghosts unnecessary. To incorporate gravity into quantum field theory, we introduce an extended set of 256×256 gamma matrices. We further propose a phase–transition mechanism in which electrons exhibit bosonic features inside atoms while remaining fermionic outside, whereas photons display the opposite behavior. This statistical transition suggests a unified understanding of gauge theories and gravity driven by energy–dependent changes of field statistics. Keywords: phase transition; statistical inversion; quantum field theory; gauge theory; quantum gravity; duality principle; extended Dirac equation; gamma matrices 1 Contents 1 Introduction 4 1.1 Status and challenges of quantum field theory ................ 4 1.2 Our approach .................................. 5 1.3 Organization of the paper ........................... 5 2 Basic structure of conventional quantum field theory 6 2.1 Basic structure of quantum electrodynamics (QED) ............. 6 2.1.1 Gauge–fixing term ........................... 6 2.1.2 Renormalization procedure ....................... 7 2.2 Basic structure of quantum chromodynamics (QCD) ............. 7 2.2.1 Gauge–fixing term ........................... 7 2.2.2 Faddeev–Popov ghost term ....................... 7 2.2.3 BRST symmetry ............................ 8 2.3 Challenges and limitations of the conventional framework .......... 8 3 Separation of Spin and Statistics in the Dual Description 9 4 Quantum Description and Symmetry of the Metric Tensor 11 4.1 Basic Mathematical Structure ......................... 11 4.2 Comparison between the Conventional and Extended Dirac Equations . . . 12 4.3 Detailed structure of the gamma system ................... 12 4.3.1 Construction of the basic operators .................. 12 4.4 Anticommutation relations and metric structure ............... 13 4.5 The 256 ×256 bosonic gamma matrices Ω .................. 13 5 Construction of Extended QED and Extended QCD 15 5.1 Construction of extended QED ........................ 15 5.1.1 Extended Lagrangian (e-QED) .................... 15 5.2 Construction of extended QCD ........................ 15 5.2.1 Extended Lagrangian .......................... 15 5.2.2 Physical degrees of freedom of gluons ................. 16 6 Mechanism of Gravitational Field Generation 16 6.1 Review of the Extended QED Lagrangian .................. 16 6.1.1 Extended Lagrangian (e-QED) .................... 16 6.1.2 Gauge Symmetry ............................ 16 6.2 Structure of the Energy–Momentum Tensor and the Four Basis States . . . 17 6.2.1 Contribution from the Electron Field ................. 17 6.2.2 Contribution from the Photon Fields ................. 17 6.3 Generation of an Effective Gravitational Source via Transition Functions . 18 6.3.1 General Form of the Transition Functions .............. 19 6.3.2 Connection to an Effective Einstein Equation ............ 19 6.4 Phase Transition Inside Atoms and the Gravitational Field ......... 19 6.4.1 Effective Pressure Inside Atoms .................... 20 6.4.2 Analogy with the Superconducting State ............... 20 6.4.3 Origin of Attractive Interaction and the Four Basis States ..... 21 6.5 Quantum Generation of the Gravitational Field ............... 22 2 6.5.1 Induced Gravitational Action ..................... 22 6.5.2 Gravitational Potential in the Weak-field Approximation . . . . . . 23 6.6 Physical Consequences and Observational Prospects ............. 23 6.6.1 Gravitational Effects Inside Atoms .................. 23 6.6.2 Gravitational Effects in Superconductors ............... 24 6.6.3 High-energy Limit and UV Divergence Avoidance .......... 24 6.7 Theoretical Implications and Future Prospects ................ 25 6.8 Summary of This Section ............................ 26 7 Conclusion 26 8 Acknowledgments 31 A Detailed Computations of Gamma Matrices 34 A.1 Basic 4 ×4 gamma matrices .......................... 34 A.2 Explicit 256 ×256 extension .......................... 34 A.3 Verification of anticommutation relations ................... 35 B On the Physical Interpretation of the 256-Dimensional Extension 35 C Proof of Unitarity of the Scattering Matrix 36 C.1 General properties of the S–matrix ...................... 36 C.2 Verification of the Ward–Takahashi identity ................. 36 C.3 Direct proof of unitarity ............................ 37 C.4 Check of gauge invariance ........................... 37 D Proof of Gauge–Fixing–Free Formulation 37 D.0.1 Step 1: Automatic restriction of degrees of freedom ......... 38 D.0.2 Step 2: Preservation of gauge invariance ............... 38 D.0.3 Step 3: Proof of unitarity ....................... 38 E Path Integral for QED Without Gauge Fixing and Ghosts 38 E.1 Theoretical framework ............................. 38 E.1.1 Conventional QED path integral .................... 38 E.1.2 New formulation ............................ 39 F Reference Formulas of the Standard Model and the Present Extension (e–SM) 39 F.1 Reference formulas of the Standard Model (SM) ............... 39 F.2 Notation of the present framework (effective gamma) and replacement rules 40 F.3 Extended Standard Model (e–SM) in the present framework ........ 40 F.4 Consistency in limits and replacement summary ............... 41 F.5 Coupling to gravity (master action) and ghost–free property ........ 41 G Concrete Example: Compton Scattering 42 G.0.1 Calculation in Minkowski spacetime .................. 42 G.1 Metric tensor and extended gamma matrices ................. 42 G.1.1 Application of the extended gammas to Compton scattering . . . . 42 G.1.2 Concrete impact of the metric tensor on Compton scattering . . . . 43 G.1.3 Explicit modifications in the Compton calculation .......... 43 3 G.1.4 Comparison between Minkowski and curved–spacetime results . . . 44 H Obtaining the Mathematica Codes 45 H.1 Code structure ................................. 45 I Data Availability 45 1 Introduction Quantum field theory (QFT) is one of the most important achievements of twentieth–century physics. It serves as the foundational framework of particle physics and provides the most successful theoretical description of the fundamental interactions in nature. In particular, quantum electrodynamics (QED) agrees with experiments with astonishing precision [1,2,3], and quantum chromodynamics (QCD) successfully explains the essential properties of the strong interaction [4,5,6]. The issue of statistics in QFT remains an active topic of research [7,8]. In particular, the behavior of statistics in the high–energy limit has attracted attention in connection with quantum gravity [9]. Moreover, observations of analogous phenomena in condensed–matter physics [10] suggest the possibility of testing predictions of such theories. Nevertheless, several fundamental challenges remain in the standard formulation of QFT: 1.1 Status and challenges of quantum field theory First, the introduction of gauge fixing and ghost fields―required for the quantization of gauge fields―obscures the physical interpretation of the theory. For example, in QCD one must impose gauge–fixing conditions for each of the eight gluon fields and introduce the corresponding ghost fields. This substantially complicates the structure of the theory and hinders physical interpretation. Second, a unified description of fields with different statistics (fermions and bosons) has not been achieved. Supersymmetry offers one possible answer to this problem, but it has not yet been verified experimentally. Furthermore, supersymmetry requires partner particles for each particle, which increases the complexity of the theory. Third, it is difficult to couple the theory consistently to gravity. Quantization of gravity based on general relativity faces ultraviolet (UV) divergences that cannot be resolved within the usual field–theoretic framework. This poses a serious challenge to the consistency of the theory in the high–energy limit. Behind these issues lie the following basic questions: 1. Can field statistics vary with the energy scale? •At low energies, electrons behave as fermions and obey the Pauli exclusion principle. •However, this picture may change at ultra–high energies. •In superconductivity, electrons effectively exhibit bosonic behavior through Cooper–pair formation. 4 •It is necessary to examine whether such statistical changes can occur at a more fundamental level. 2. What is the relation between the gauge principle and gravity? •Gauge fields and the gravitational field appear to be based on different symmetries. •Yet the two may be fundamentally connected. •A unified description at high energies is required. 3. Do ultraviolet divergences carry physical meaning? •Current theories require artificial regularization. •This may indicate an incompleteness of the framework. •A more fundamental theory might resolve these issues naturally. 1.2 Our approach As a new approach to these challenges, we propose a fermion–boson duality. This duality is formulated mathematically through an extended Dirac equation and has the following features: 1. It indicates that field statistics can change dynamically with the energy scale. 2. It enables a formulation that does not require gauge fixing or ghost fields. 3. It provides a mechanism by which the gravitational field emerges naturally as a quantum effect. Specifically, we introduce an extended set of 256 ×256 gamma matrices and use them to generalize the conventional four–dimensional Dirac equation. This extension offers the following advantages: 1. A unified description of fermionic and bosonic degrees of freedom. 2. A natural realization of the physical degrees of freedom of gauge fields (two transverse polarizations). 3. A potential route toward the automatic avoidance of ultraviolet divergences. 1.3 Organization of the paper The paper is organized as follows. We first provide a detailed overview of the basic structure of conventional QFT and its challenges, focusing in particular on the need for gauge fixing and ghost fields, the problem of ultraviolet divergences, and the difficulties of coupling to gravity. We then introduce bosonic and fermionic gamma matrices and the duality framework, and develop its basic mathematical structure. We construct the extended 256 ×256 gamma matrices and formulate the extended Dirac equation based on them. Subsequently, we discuss a quantum description of the metric tensor and its symmetries, indicating a natural coupling to the gravitational field. We then examine concrete applications of the present framework: 5 •Construction of extended quantum electrodynamics (e-QED) •Construction of extended quantum chromodynamics (e-QCD) •Mechanism for the emergence of the gravitational field 2 Basic structure of conventional quantum field theory QFT is built on the basic idea of field quantization. In this section we first review the basic structure of the conventional theory, and then discuss in detail the challenges it currently faces. 2.1 Basic structure of quantum electrodynamics (QED) QED is the most successful quantum field theory describing electromagnetic interactions. Its basic Lagrangian is LQED =ψ(iγµDµ−m)ψ−1 4FµνFµν.(1) This Lagrangian consists of the following three basic terms: 1. The first term is the kinetic term of the Dirac field, describing the free motion of a charged particle and its minimal coupling to the electromagnetic field, with Dµ=∂µ+ieAµ. 2. The second term is the mass term characterizing the fermion mass. 3. The third term is the kinetic term of the electromagnetic field, including the field–strength tensor Fµν =∂µAν−∂νAµ. The theory is invariant under the following gauge transformations: ψ→e−ieα(x)ψ, (2a) Aµ→Aµ+∂µα(x).(2b) This gauge invariance is a fundamental symmetry of the theory and leads to charge conservation. However, the structure involves two important technical challenges: 1. Infrared divergences in the photon propagator; 2. Ultraviolet divergences in perturbation theory. To address these problems, the conventional theory employs the following techniques. 2.1.1 Gauge–fixing term The gauge–fixing term added to the QED Lagrangian takes the form LGF =−1 2ξ(∂µAµ)2,(3) where ξis the gauge parameter (in the Lorenz gauge one typically takes ξ= 1). This term allows a proper definition of the photon propagator and enables path–integral computations. 6 2.1.2 Renormalization procedure To treat divergent terms appropriately, one introduces the renormalized Lagrangian Lren =Z2ψ(iγµDµ−m)ψ−Z3 4FµνFµν,(4) where Z2and Z3are the renormalization factors for the fermion and the photon fields, respectively. By choosing these factors appropriately, the finiteness of the theory is ensured. This is the standard QED prescription and yields results in remarkable agreement with experiments. 2.2 Basic structure of quantum chromodynamics (QCD) QCD is a non–Abelian gauge theory describing the strong interaction. Its basic Lagrangian is LQCD = nf X f=1 qf(iγµDµ−mf)qf−1 4Ga µνGaµν,(5) with the covariant derivative Dµ=∂µ+igsTaGa µ.(6) A characteristic structure of the theory appears in the non–Abelian field–strength tensor [11]: Ga µν =∂µGa ν−∂νGa µ+gsfabcGb µGc ν.(7) Because of non–Abelianity, QCD faces the following technical issues: 1. The need for gauge fixing to remove redundant degrees of freedom of the gluon fields; 2. Introduction of Faddeev–Popov ghost fields [12] to cancel unphysical modes; 3. Realization of asymptotic freedom―i.e., the decrease of the coupling at high energies [4,5]. To address these, the conventional framework introduces the following terms. 2.2.1 Gauge–fixing term To control the longitudinal components of the gluon fields Ga µ, LGF =−1 2ξ(∂µGaµ)2(8) is added, where ξis the gauge parameter (one typically takes ξ= 1 in the Lorenz gauge). 2.2.2 Faddeev–Popov ghost term To ensure unitarity in non–Abelian gauge theories, one introduces Lghost =ca∂µDµca, Dµca=∂µca+gfabcGbµcc,(9) where ca, caare the Faddeev–Popov ghosts, fabc are the structure constants, and gis the coupling constant. 7 2.2.3 BRST symmetry The full theory including the gauge–fixing and ghost terms is invariant under the BRST transformations [13]:[14] δBGa µ=Dµca,(10a) δBca=−g 2fabccbcc,(10b) δBca=1 ξ∂µGa µ.(10c) This symmetry guarantees the physical unitarity of the theory. 2.3 Challenges and limitations of the conventional framework The conventional formulation of QFT faces the following basic challenges: 1. Consistency with gravity •Perturbative non–renormalizability; •Background dependence; •Hierarchies of energy scales. 2. Technical complexity of the mathematical structure •Arbitrariness of gauge fixing; •Introduction of ghost degrees of freedom; •Complicated treatment of divergences. 3. Opacity of physical interpretation •Physical meaning of virtual processes; •Presence of unphysical degrees of freedom; •Complexity of the vacuum structure. These challenges suggest the need for a more fundamental theory. Program of this paper and lead-in to the next section. The issues listed in the previous section―(1) consistency with gravity, (2) mathematical complexity, and (3) opacity of physical interpretation―partly stem from the conventional framework treating the spin representation and statistics as inseparable. In this paper we aim to address them with a design that satisfies the following minimal requirements: (P1) While preserving four-dimensional Lorentz symmetry and locality, describe the theory with a factorized spin–statistics structure. (P2) Treat field statistics as a continuous quantity depending on energy (or effective density), and recover the standard theory exactly in the low-energy limit. (P3) For gauge fields, use an Ω-type operator to automatically select only the two physical transverse components, thereby avoiding gauge fixing and the introduction of ghosts. 8 (P4) Introduce a statistical-duality weighting into the gauge kinetic terms so that an effective gravitational source naturally emerges from the variational principle. As a minimal implementation of this guideline, each field X∈ {e, γ}is extended to two components {XF, XB}, and the electron and photon are represented in the four-state basis {eF, eB, γF, γB}. We further introduce a transition function TXF(E)∈[0,1] so that, in the low-energy limit, (TeF, TγF)→(1,0) (the standard Fermi/Bose statistics), while at high energies or in high-density regions the weights can deviate smoothly from (1,0). This construction appears in the subsequent e-QED/e-QCD as a two-term mixture such as (1−T)F2+T T2 (field), and by replacing the right-hand side of the Einstein equation with a convex combination of Θ(A) µν and Θ(B) µν , it enables us to describe the contribution of induced gravity. Based on these considerations, in the next section we present, under minimal assumptions, the definition of the four-state basis and the transition rule. 3 Separation of Spin and Statistics in the Dual Description Four basis states and energy–dependent mixing We decompose the quantum state of the electron and the photon as |ψtotali=|ψeFi+|ψeBi+|ψγFi+|ψγBi,(11) and introduce an energy–dependent transition function T(E)∈[0,1] so that |ψe(E)i=TeF(E)|ψeFi+1−TeF(E)|ψeBi,(12) |ψγ(E)i=TγF(E)|ψγFi+1−TγF(E)|ψγBi,(13) so that the weights sum to unity in each line. In practice one may use a logistic form TXF(E) = 1 1 + exp (E−Efb)/(ℏν), which reproduces the standard statistics (electron = Fermi, photon = Bose) for EEfb and inverts the statistics for EEfb. Physical implications (summary) (i) At low energies the framework reduces to standard QED/QCD. (ii) In the transition region, fermionic and bosonic components can coexist. (iii) At high energies, the dual side dominates, enabling a description of UV divergence suppression and an enhanced effective gravitational source. 9 5.2.2 Physical degrees of freedom of gluons In standard QCD, the gluon fields Ga µcarry 4 ×8 = 32 degrees of freedom; to restrict them to the physical 2 transverse polarizations ×8 colors = 16 degrees, one must impose gauge fixing and introduce ghosts. In the present framework: 1. The bosonic gamma Ωb νautomatically removes the longitudinal components, leaving the two transverse modes. 2. The non-Abelian structure is retained while only physical degrees of freedom survive. 3. The framework suggests the possibility of new physical effects in the strong–coupling regime. 6 Mechanism of Gravitational Field Generation In this section, we discuss the mechanism by which the gravitational field naturally emerges as a quantum effect within the fermion–boson duality theory [17, 18, 19, 15]. We formulate how the effective energy–momentum tensor generated through statistical transitions affects the spacetime metric, and clarify the relation between phase transitions inside atoms and the gravitational field. 6.1 Review of the Extended QED Lagrangian Before discussing the mechanism of gravitational field generation, we briefly review the Lagrangian of extended quantum electrodynamics (e-QED) [1, 20, 21, 22, 15]. 6.1.1 Extended Lagrangian (e-QED) LextQED =ψhi(Γbµ+ Ωbµ)Dµ−miψ−1 4FµνFµν −1 4TµνTµν,(30) where Γbµ=gµνΓb ν, Ωbµ=gµνΩb ν, and the covariant derivative is Dµ=∂µ+ie Aµ+ie′Bµ,(31) while the field strengths are Fµν =∂µAν−∂νAµ, Tµν =∂µBν−∂νBµ.(32) Here Aµdenotes the usual bosonic photon field, Bµthe fermionic photon field, and eand e′ are the corresponding coupling constants. (The original expression i gµνΓνDµ+i gµν ΩνDµ is equivalent to (30) up to dummy index relabeling [15].) 6.1.2 Gauge Symmetry This Lagrangian is invariant under the following gauge transformations [11, 12, 13, 14]: ψ→e−i(eα+e′β)ψ, (33a) Aµ→Aµ+∂µα, (33b) Bµ→Bµ+∂µβ, (33c) where α(x) and β(x) are arbitrary real functions. 16 6.2 Structure of the Energy–Momentum Tensor and the Four Basis States In the extended quantum field theory, the energy–momentum tensor receives contributions from all four basis states introduced in the previous section [21, 18]. Its general form is Θµν = Θ(eF) µν + Θ(eB) µν + Θ(γB) µν + Θ(γF) µν ,(34) where •Θ(eF) µν : contribution from the fermionic electron field (associated with Γbµ), •Θ(eB) µν : contribution from the bosonic electron field (associated with Ωbµ), •Θ(γB) µν : contribution from the bosonic photon field Aµ, •Θ(γF) µν : contribution from the fermionic photon field Bµ. This structure makes explicit how the four basis states {eF, eB, γF, γB}, shown in Fig. 1 [15], participate in the generation of the gravitational field. 6.2.1 Contribution from the Electron Field The contribution from the electron field splits into two parts, originating from the Γbµ term and the Ωbµterm in the extended Dirac equation (30). Contribution from the Fermionic Electron Field The part associated with Γbµis Θ(eF) µν =i 4hψΓb µDνψ+ψΓb νDµψ−(Dµψ)Γb νψ−(Dνψ)Γb µψi−gµνLΓ,(35) where LΓ=ψ[iΓbµDµ−m]ψis the Lagrangian density of the fermionic electron sector. Contribution from the Bosonic Electron Field The part associated with Ωbµis Θ(eB) µν =i 4hψΩb µDνψ+ψΩb νDµψ−(Dµψ)Ωb νψ−(Dνψ)Ωb µψi−gµνLΩ,(36) where LΩ=ψ[iΩbµDµ]ψis the Lagrangian density of the bosonic electron sector (the mass term is contained only in the fermionic component). The total contribution from the electron field is Θ(e) µν = Θ(eF) µν + Θ(eB) µν .(37) Here Dµ=∂µ+ie Aµ+ie′Bµis the covariant derivative in (31). 6.2.2 Contribution from the Photon Fields The contributions from the photon sector split into two parts, coming from the bosonic photon Aµand the fermionic photon Bµ. 17 Contribution from the Bosonic Photon Field Aµ(γBcomponent) Θ(γB) µν =FµλFλ ν−1 4gµνFλρFλρ,(38) where Fµν =∂µAν−∂νAµis the usual electromagnetic field strength. This term describes the standard low-energy contribution from the photon field [1, 20]. Contribution from the Fermionic Photon Field Bµ(γFcomponent) Θ(γF) µν =TµλTλ ν−1 4gµνTλρTλρ,(39) where Tµν =∂µBν−∂νBµis the field strength of the fermionic photon field. This contribution becomes important in high-energy or high-density regimes [15]. The total contribution from the photon fields is then Θ(γ) µν = Θ(γB) µν + Θ(γF) µν .(40) 6.3 Generation of an Effective Gravitational Source via Transition Functions Because the transition functions TeF(E) and TγF(E) depend on energy, the contributions of the four basis states are dynamically weighted [15, 23]. The effective energy–momentum tensor takes the form hΘµνi=TeFhΘ(eF) µν i+ (1 −TeF)hΘ(eB) µν i + (1 −TγF)hΘ(γB) µν i+TγFhΘ(γF) µν i.(41) This expression exhibits the following energy-dependent behavior: Low-energy limit (EEfb) TeF→1, TγF→0,(42) so that hΘµνi → hΘ(eF) µν i+hΘ(γB) µν i,(43) and the fermionic electron and bosonic photon dominate. This reproduces the standard picture of quantum electrodynamics [1, 20]. High-energy limit (EEfb) or high-density regime TeF→0, TγF→1,(44) so that hΘµνi → hΘ(eB) µν i+hΘ(γF) µν i,(45) and the bosonic electron and fermionic photon dominate. This corresponds to the statistically inverted state. 18 Transition region (E∼Efb)In an intermediate energy regime such as the interior of atoms, all four components acquire finite weights and a mixed quantum-statistical state is realized. This transition region plays a central role in the generation of the gravitational field [15]. 6.3.1 General Form of the Transition Functions We adopt a logistic form for the transition functions [15]: TXF(E) = 1 1 + exp(E−Efb)/(ℏν),(46) where Efb is the transition energy and ℏνcharacterizes the sharpness of the transition. In the low-energy limit (EEfb) we have TeF→1, TγF→0,(47) so that the theory reduces to the standard quantum field theory [21, 22]. In the high-energy limit (EEfb) or a high-density regime we find TeF→0, TγF→1,(48) and a statistical inversion takes place. 6.3.2 Connection to an Effective Einstein Equation The dynamics of the spacetime metric gµν are governed by an effective Einstein equation [18, 24, 19]: Gµν + Λgµν =8πG c4hΘµνi,(49) where Gµν is the Einstein tensor, Λ the cosmological constant, and GNewton’s gravitational constant. When transition functions are present, the right-hand side can be written, using (41), as hΘµνi=hΘ(F) µν i+ (1 −hTγFi)hΘ(A) µν i+hTγFihΘ(B) µν i.(50) This structure shows that the relative weights of the bosonic photon field Aµand the fermionic photon field Bµvary with the energy scale, and that this variation directly contributes to the curvature of spacetime [17, 18]. 6.4 Phase Transition Inside Atoms and the Gravitational Field Inside atoms, electrons move under the strong Coulomb potential of the nucleus, leading to an effectively ultra-high pressure and high-density environment. In such a situation, a phase transition in the statistics of electrons may occur [1, 25, 26]. 19 6.4.1 Effective Pressure Inside Atoms The electric field energy density uEnear the nucleus, from classical electromagnetism, is uE=ϵ0 2E2∼1 2ϵ0Ze 4πϵ0r22 ∼Z2e2 32π2ϵ0r4,(51) where Zis the atomic number, ethe elementary charge, and rthe distance from the nucleus [1, 25]. This relation holds quite generally, independently of the specific theory. In the region near the Bohr radius a0≈0.53 ×10−10 m, this electric field energy density is extremely large, corresponding to an effective pressure Peff ∼uE∼1020 Pa,(52) which is about 108times the pressure at the Earth’s center (∼3.6×1011 Pa). The distinctive feature of the present theory is the interpretation that, in this generally accepted ultra-high-pressure environment, a statistical phase transition is triggered and the transition functions change as (TeF, TγF) : (1,0) →(0,1), thereby generating a gravitational field. 6.4.2 Analogy with the Superconducting State This ultra-high-pressure environment is analogous to that in which Cooper pairs form in superconductors. In a superconducting state, electrons form Cooper pairs and exhibit effectively bosonic behavior [27, 26, 28, 29]. As shown in the attached figure (Fig. 2), the following phase transition process is expected to occur inside atoms: Fermionic electron (massive) Bosonic electron (massless) Bosonic electron (massless) Fermionic photon (massive) Repulsion Attraction Normal Superconducting state (this is the true nature of gravity) Undergoes a phase transition inside an atom Fermionic electron (massive) Bosonic electron (massless) Figure 2: Statistical phase transition inside atoms and the nature of gravity. Left: In the normal state, fermionic electrons exhibit repulsive interactions due to the Pauli exclusion principle. Right: Under the ultra-high-pressure environment inside atoms, electrons acquire bosonic properties, while photons acquire fermionic properties, and a phase transition takes place. This statistical inversion leads to an effective attractive interaction between electrons and the emergence of a superconducting-like state. The picture suggests that this phenomenon underlies the true nature of gravity [27,26,28]. 20 Details of the Phase Transition Mechanism As illustrated in Fig. 2, the statistical phase transition between the inside and outside of the atom can be understood as a change in the weights of the four basis states [15]: 1. Normal state (outside the atom, EEfb): •Electrons: fermionic (eF, massive, TeF≈1) •Photons: bosonic (γB, massless, TγF≈0) •Electron–electron interaction: repulsive (Pauli exclusion principle) •Energy–momentum tensor: hΘµν i ≈ hΘ(eF) µν i+hΘ(γB) µν i 2. Transition region (inside the atom, E∼Efb): Under the ultra-high pressure Peff ∼1020 Pa, the transition functions take intermediate values: 0< TeF<1,0< TγF<1.(53) In this regime, all four basis states have finite weights and a mixed statistical state is realized: hΘµνi=X X∈{eF,eB,γF,γB} wX(E)hΘ(X) µν i,(54) where wX(E) denotes the weight of each basis state. 3. Statistically inverted state (deep inside the atom or at high energy, E Efb): •Electrons: dominantly bosonic (eB, effectively reduced mass, TeF≈0) •Photons: acquire fermionic properties (γF, effectively massive, TγF≈1) •Electron–electron interaction: attractive (analogous to Cooper-pair formation) [27, 26] •Energy–momentum tensor: hΘµν i ≈ hΘ(eB) µν i+hΘ(γF) µν i The crucial point is that, during this transition, all four basis states {eF, eB, γF, γB} undergo dynamic changes in their weights depending on the energy scale [15]. 6.4.3 Origin of Attractive Interaction and the Four Basis States The statistical inversion modifies the effective interaction between electrons. The interaction potential between bosonic electrons eBtakes the form VeB−eB(r)∝ −(1 −TeF)2αℏc r,(55) which is attractive. Here αis the fine-structure constant [1]. 21 This attractive interaction is related to gravity through the following four-step mechanism [17, 15]: statistical inversion (TeF: 1 →0, TγF: 0 →1) ↓ change in the weights of the four basis states ({eF, γB} → {eB, γF}) ↓ emergence of attractive interaction (VeB−eB<0) ↓ redistribution of energy–momentum (change in the components of Θµν) ↓ distortion of spacetime (Gµν ∝Θµν) (56) An especially important point is that each component of the energy–momentum tensor changes dynamically via the transition functions: ∆Θµν =hTeFΘ(eF) µν + (1 −TeF)Θ(eB) µν i +h(1 −TγF)Θ(γB) µν +TγFΘ(γF) µν i−Θ(standard) µν , (57) where Θ(standard) µν = Θ(eF) µν + Θ(γB) µν is the energy–momentum tensor of standard quantum electrodynamics [1, 20]. It is this ∆Θµν that serves as the source of the gravitational field. 6.5 Quantum Generation of the Gravitational Field 6.5.1 Induced Gravitational Action The energy–momentum tensor containing the transition functions induces the following effective action for the spacetime metric [17]: Sinduced =Zd4x√−ghc4 16πGR+Lmatter +Ltransi,(58) where Ltrans is the additional term associated with the transition: Ltrans =−1 2∂µTXF∂µTXF+V(TXF).(59) The potential V(TXF) encodes the stability of the transition. Typically, it takes a Mexican-hat form: V(TXF) = λh(TXF−T0)2−v2i2,(60) where T0= 1/2 is the symmetric point, va vacuum expectation value, and λa coupling constant. 22 6.5.2 Gravitational Potential in the Weak-field Approximation In the weak-field approximation gµν =ηµν +hµν with |hµν |  1, linearizing the Einstein equation yields □hµν =−16πG c4Θµν −1 2ηµνΘλ λ,(61) where □=ηµν∂µ∂νis the d’Alembertian [18, 19]. For a static, spherically symmetric configuration, taking h00 =−2Φ/c2gives ∇2Φ = 4πGρeff ,(62) where the effective mass density is ρeff =ρ0+ ∆ρ(e) trans + ∆ρ(γ) trans.(63) Here the correction from the electron sector is ∆ρ(e) trans =1 c2hhΘ(eB) 00 i−hΘ(eF) 00 ii(1 −TeF),(64) and the photon-sector correction is ∆ρ(γ) trans =1 c2hhΘ(γF) 00 i−hΘ(γB) 00 iiTγF.(65) These corrections have the following physical meaning: •∆ρ(e) trans: change in the effective mass density due to bosonization of electrons, •∆ρ(γ) trans: change in the effective mass density due to fermionization of photons. Through statistical transitions, these corrections become nonzero and directly affect the gravitational field [17, 15]. 6.6 Physical Consequences and Observational Prospects 6.6.1 Gravitational Effects Inside Atoms Inside atoms (r∼a0), the gravitational correction induced by the transition can become non-negligible. The relative correction is estimated as ∆Φ ΦNewton ∼∆ρtrans ρ0∼TγF(Eatom),(66) where Eatom ∼10 eV is a typical atomic energy scale. Assuming a transition energy Efb ∼1 MeV [15], we find TγF(10 eV) ≈1 1 + exp[(10 eV −1 MeV)/(100 keV)] ≈1,(67) showing that the statistical inversion is essentially completed inside atoms. 23 6.6.2 Gravitational Effects in Superconductors In the superconducting state, electrons form Cooper pairs and behave effectively as bosons [27, 26, 28]. This corresponds, in the present theory, to the limit TeF→0. The effective gravitational constant inside a superconductor may be written as Geff =Gh1 + αsc(1 −TeF)2i,(68) where αsc is a coupling constant characteristic of the superconducting state. Due to the Meissner effect, the electromagnetic field inside the superconductor acquires an effective mass mγ,eff =ℏ λL ,(69) where λLis the London penetration depth [29, 28]. Within our framework, this phenomenon can be interpreted as follows: •Inside the superconductor, TγF≈1 so that the fermionic photon field Bµbecomes dominant. •Unlike the bosonic photon field Aµ, the fermionic photon field can acquire an effective mass. •This provides an explanation of the Meissner effect as the expulsion of magnetic fields [29, 28]. Thus, in vacuum TγF≈0 and the massless bosonic photon Aµis dominant. Inside a superconductor, however, TγF≈1 and the massive fermionic photon field Bµdominates, offering a microscopic mechanism for the Meissner effect. 6.6.3 High-energy Limit and UV Divergence Avoidance In the high-energy limit (EEfb), the exponential fall-off of the transition function leads to a natural suppression of ultraviolet divergences [20, 21, 15]. For example, the one-loop self-energy correction of the electron is Σ(p) = Zd4k (2π)4[TeF(k)]3e2 k2 1 / p−/ k−m,(70) and because [TeF(k)]3∼exp[−3(k−Efb)/(ℏν)], the contributions from kEfb are exponentially suppressed, rendering the integral finite. Numerically, it has been found [15] that Σreg ≈0.0134 e2 Efb ,(71) which is a finite value. 24 6.7 Theoretical Implications and Future Prospects The mechanism of gravitational field generation presented in this section yields several important implications: 1. Quantum origin of gravity: Gravity may not be a fundamental interaction. Instead, it may be an induced effect arising from statistical transitions among the four basis states {eF, eB, γF, γB}[15]. This can be understood as a statistical–mechanical realization of Sakharov s induced gravity [17]. 2. Unification of gauge principle and gravity: Since statistical transitions of the gauge fields (Aµand Bµ) directly generate spacetime curvature, this framework naturally unifies gauge theory and gravity [18, 7, 8, 9]. In particular, all four basis states contribute to the metric through the energy–momentum tensor. 3. Candidates for dark matter: •Bosonic electron eB: In ordinary detectors, only eFis effectively observed, but in high-density regions the eBcomponent could contribute as a gravitational source. •Fermionic photon γF: Normally observed as γB, but under extreme conditions the γFcomponent may acquire a mass and become a candidate for dark matter. These components are suppressed by the transition functions and hence are difficult to detect in standard observations [30]. 4. Cosmological implications: In the early Universe, where the energy scale was very high, the statistical inversion would have been dominant, and the energy–momentum tensor would have been mainly determined by hΘ(eB) µν i+hΘ(γF) µν i. This may have influenced the formation of large-scale structure in the Universe [18, 10]. 5. Testability of the four basis states: Although it is in principle difficult to separate the contributions from each basis state, the following indirect tests are conceivable: •Measuring the forms of the transition functions TeF(E) and TγF(E) [15], •Detecting tiny corrections to atomic energy levels due to interference among the four components, •Searching for signatures of statistical inversion in high-energy collision experiments [20, 21], •Looking for gravitational anomalies inside superconductors due to the contributions of eBand γF[27, 26, 28]. 6. Possibility of experimental verification: •Precision measurements of gravity inside superconductors (detection of ∆ρ(e) trans+ ∆ρ(γ) trans), •Precision spectroscopy of atomic energy levels [1, 25], •Searches for statistical inversion effects in high-energy experiments [20, 21], •Observation of interference effects among the four basis states [15]. 25 [3] Kinoshita, T., “Theoretical Overview of the Electron and Muon Anomalous Magnetic Moments,” Rev. Mod. Phys. 81, 865–934 (2009). [4] Gross, D. J., “Ultraviolet Behavior of Non-Abelian Gauge Theories,” Phys. Rev. Lett. 30, 1343–1346 (1973). [5] Politzer, H. D., “Reliable Perturbative Results for Strong Interactions?,” Phys. Rev. Lett. 31, 581–583 (1973). [6] Wilczek, F., “Asymptotic Freedom in Parton Language,” Phys. Rev. Lett. 31, 621–623 (1973). [7] Witten, E., “Topological Quantum Field Theory,” Commun. Math. Phys. 174, 155–186 (1995). [8] Gaiotto, D., “N= 2 Dualities,” Adv. Theor. Math. Phys. 19, 1163–1205 (2015). [arXiv:1412.3478] [9] Strominger, A., “Black Hole Soft Hair,” JHEP 1401, 034 (2014). [arXiv:1401.0347] [10] Read, N., “Non-Abelian Statistics and the Fractional Quantum Hall Effect,” Rev. Mod. Phys. 73, 913–1021 (2001). [arXiv:cond-mat/9909133] [11] Yang, C. N., Mills, R. L., “Conservation of Isotopic Spin and Isotopic Gauge Invariance,” Phys. Rev. 96, 191–195 (1954). [12] Faddeev, L. D., Popov, V. N., “Feynman Diagrams for the Yang–Mills Field,” Phys. Lett. B 25, 29–30 (1967). [13] Becchi, C., Rouet, A., Stora, R., “Renormalization of Gauge Theories,” Ann. Phys. 98, 287–321 (1975). [14] Kawamura, Y., Gauge Theory from Basic Physics (in Japanese), Saiensu-sha, 2017, ASIN: B0785MQP9Q. [15] H. Maruyama, “Proposal for Statistical Mechanics-Based UV Regularization Using Fermion–Boson Transition Functions,” Frontiers in Physics 13, 1618853 (2025). doi:10.3389/fphy.2025.1618853. [16] Sato, H., Groups and Physics (in Japanese), Maruzen Publishing, 2016, ISBN: 9784621300848. [17] Sakharov, A. D., “Vacuum Quantum Fluctuations in Curved Space and the Theory of Gravitation,” Dokl. Akad. Nauk SSSR 177, 70–71 (1967) [ Sov. Phys. Dokl. 12, 1040–1041 (1968) ]. [18] Weinberg, S., Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, 1972. [19] Wald, R. M., General Relativity, University of Chicago Press, 1984. [20] Peskin, M. E., Schroeder, D. V., An Introduction to Quantum Field Theory, Addison–Wesley, 1995. 32 [21] Weinberg, S., The Quantum Theory of Fields, Vol. 1: Foundations, Cambridge University Press, 1995. [22] Sakamoto, M., Quantum Field Theory (II): Feynman Diagrams and Renormalization (in Japanese), Shokabo, 2020, ISBN-10: 4785325127, ISBN-13: 978-4785325121. [23] H. Maruyama, “Dirac Operator in Curved Spacetime via a Clifford-Algebraic 256 ×256 Matrix Representation and Applications to QED Processes,” Frontiers in Physics, 2025. [24] Misner, C. W., Thorne, K. S., Wheeler, J. A., Gravitation, W. H. Freeman, 1973. [25] Sakurai, J. J., Modern Quantum Mechanics, Revised Edition, Addison–Wesley, 1994. [26] Chen, Q., Levin, K., Chien, C.-C., He, Y., “Comparison of Different Pairing Fluctuation Approaches to BCS–BEC Crossover,” Phys. Rep. 412, 1–88 (2005). arXiv:0810.1938. [27] Bardeen, J., Cooper, L. N., Schrieffer, J. R., “Theory of Superconductivity,” Phys. Rev. 108, 1175–1204 (1957). [28] Tinkham, M., Introduction to Superconductivity, 2nd ed., McGraw–Hill, 1996. [29] London, F., Superfluids, Vol. I: Macroscopic Theory of Superconductivity, Wiley, 1950. [30] Bertone, G., Hooper, D., Silk, J., “Particle Dark Matter: Evidence, Candidates and Constraints,” Phys. Rep. 405, 279–390 (2005). [31] Dirac, P. A. M., “The Quantum Theory of the Electron,” Proc. R. Soc. Lond. A 117, 610–624 (1928). [32] Klein, O., Nishina, Y., “On the Scattering of Radiation by Free Electrons According to Dirac’s New Relativistic Quantum Mechanics,” Z. Phys. 52, 853–868 (1929). 33 A Detailed Computations of Gamma Matrices A.1 Basic 4×4gamma matrices We start from a complete representation of the four–dimensional gamma matrices: γ0=    1 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 −1    (81a) γ1=i    0001 0010 0100 1000    (81b) γ2=i    000−i 0 0 i0 0−i0 0 i000    (81c) γ3=i    0 0 1 0 0 0 0 −1 −1 0 0 0 0 1 0 0    (81d) These matrices satisfy the anticommutation relations {γµ, γν}= 2gµνI4.(82) A.2 Explicit 256 ×256 extension The 256 ×256 extension is constructed as an eightfold tensor product of Pauli matrices. First, the basic Pauli matrices: σx=0 1 1 0, σy=0−i i0,(83a) σz=1 0 0−1, e =1 0 0 1.(83b) Using these, we build the 256 ×256 extended gamma matrices, e.g., Γ0=σy⊗σz⊗σz⊗σz⊗σz⊗σz⊗σz⊗σz Γ1=−σx⊗σz⊗σz⊗σz⊗σz⊗σz⊗σz⊗σz Γ2=e⊗σy⊗σz⊗σz⊗σz⊗σz⊗σz⊗σz . . . Γ15 =−e⊗e⊗e⊗e⊗e⊗e⊗e⊗σx.(84) 34 A.3 Verification of anticommutation relations We compute the anticommutators of these matrices explicitly. First, recall the basic identities: {σi, σj}= 2δijI2(i, j =x, y, z),(85a) [σi, e] = 0.(85b) Together with the tensor–product property (A⊗B)(C⊗D) = (AC)⊗(BD),(86) we can evaluate the anticommutators of the Γ matrices. As an example, for {Γ0,Γ1}: {Γ0,Γ1}= Γ0Γ1+ Γ1Γ0 = (σy⊗σz⊗···)(−σx⊗σz⊗···) + (−σx⊗σz⊗···)(σy⊗σz⊗···) =−{σy, σx}⊗{σz, σz}⊗··· = 0.(87) Proceeding in this manner, one verifies the anticommutation relations for all combinations. B On the Physical Interpretation of the 256-Dimensional Extension One might suspect that the 256×256 matrices used in this theory imply a 256-dimensional spacetime. However, the framework remains fully consistent as a four–dimensional spacetime theory for the following reasons: 1. Preservation of physical degrees of freedom •The extended gamma matrices serve to describe internal degrees of freedom. •The actual spacetime dimensionality remains four. •The metric tensor gµν is defined on four–dimensional spacetime. 2. Consistency with observables •As demonstrated by the Mathematica computations, standard experimental results (e.g., the Klein–Nishina formula) are reproduced exactly. •If genuinely 256-dimensional effects were present, these observables would exhibit large deviations. •Agreement with four–dimensional observations indicates that the theory operates effectively in four dimensions. 35 3. Interpretation as internal structure •The 256×256 matrices are mathematical tools encoding internal statistical structure. •This is analogous to 8 ×8 matrices in SU(3) QCD, which do not increase spacetime dimensionality. •The extension enriches the mathematical structure, not the number of physical spacetime degrees of freedom. 4. Analogy with spin space •The familiar 4 ×4 Dirac matrices are compatible with four–dimensional spacetime. •Likewise, the 256 ×256 matrices are used to describe an extended spin (internal) space. •The dimensionalities of spacetime and spin/internal spaces are independent. Therefore, the use of 256 ×256 matrices in the present theory constitutes a fully consistent extension within four–dimensional spacetime physics. This is supported by the fact that all predicted physical quantities agree with observations formulated in four–dimensional spacetime. C Proof of Unitarity of the Scattering Matrix C.1 General properties of the S–matrix In the interaction picture, the scattering matrix Sis given by S=Texp−iZd4xHI(x),(88) where the interaction Hamiltonian is HI=ψigµνΓνDµ+igµν ΩµDνψ. (89) To prove unitarity, we proceed through the following steps. C.2 Verification of the Ward–Takahashi identity First, we confirm invariance under gauge transformations: ψ→e−ieα(x)ψ, (90a) Aµ→Aµ+∂µα(x).(90b) The generating functional is Z[J] = ZDψDψDAµexpiS +iZd4x JµAµ.(91) The Ward–Takahashi identity then reads ∂µ δZ[J] δJµ(x)= 0.(92) This relation is preserved even when the bosonic gamma matrices are used: ∂µhT{jµ(x)Aν(y)}i = 0.(93) 36 C.3 Direct proof of unitarity Expand the scattering operator as S= 1 + iT. (94) The unitarity condition SS†=S†S= 1 implies T−T†=i T†T. (95) To check this, consider 2→2 scattering: hf|T|ii=Mtree +Mloop,(96a) Mloop =Zd4k (2π)4M(k).(96b) With the bosonic gamma matrices, the loop integral can be written as Mloop =Zd4k (2π)4M(k) [ 1 −T(k) ],(97) and in this form one verifies the optical–theorem relation 2 Im M=X nZdΠnMnM∗ n.(98) C.4 Check of gauge invariance Finally, we confirm that gauge invariance is maintained for all processes. For the physical scattering amplitude, kµMµν =kνMµν = 0,(99) which follows naturally from the property of the bosonic gammas {Ωµ,Ων}= 2 gµν .(100) D Proof of Gauge–Fixing–Free Formulation In this framework, the absence of gauge fixing follows from a three–step argument. We work with two–index gammas and their column sums (effective gammas), consistent with the extended matrix conventions: Γµν := Γµρgρν,Γb ν:= 3 X µ=0 Γµν,Ωµν := 1 2Γµν + Γσ(µ)σ(ν),Ωb ν:= 3 X µ=0 Ωµν. {Γb µ,Γb ν}= 2gµνI256,{Ωb µ,Ωb ν}= 2g(Ω) µν I256, g(Ω) µν :=     0 0 0 0 0g11 g12 0 0g21 g22 0 0 0 0 0    ,(in flat spacetime, g(Ω) µν = diag(0,1,1,0)). 37 D.0.1 Step 1: Automatic restriction of degrees of freedom Using the bosonic effective gamma Ωbµ:= gµνΩb ν, we have ΩbµAµ(x) = ΩbµZd3p (2π)32p0aµ(p)e−ipx +a† µ(p)eipx.(101) From {Ωb µ,Ωb ν}= 2g(Ω) µν I256 it follows immediately that ΩbµAµ2=g(Ω) µν AµAν= (A1)2+ (A2)2,(102) i.e., only the two transverse components survive automatically. D.0.2 Step 2: Preservation of gauge invariance The gauge transformations δAµ=∂µα, δψ =ie α ψ (103) leave the action invariant: δS = 0.(104) (The same holds for the U(1)×U(1) extension with an additional field Bµ, upon imposing δBµ=∂µβ.) D.0.3 Step 3: Proof of unitarity The unitarity of the scattering matrix SS†=S†S= 1 (105) is satisfied without introducing gauge fixing or ghost fields. Details of the proof are given in the appendix. E Path Integral for QED Without Gauge Fixing and Ghosts In the standard path–integral (functional–integral) formulation of quantum electrodynamics (QED), gauge fixing and the introduction of Faddeev–Popov ghost fields have traditionally been required in order to remove gauge redundancy. By introducing the new mathematical structure of the bosonic gamma matrices proposed in this paper, one can formulate the theory so that only the physically necessary degrees of freedom are extracted from the outset, rendering both gauge fixing and ghosts unnecessary. E.1 Theoretical framework E.1.1 Conventional QED path integral For the standard QED Lagrangian LQED =ψiγµDµ−mψ−1 4FµνFµν,(106) 38 the photon field Aµpossesses redundancy under the gauge transformation Aµ→Aµ+∂µΛ. Hence, to define Z=ZDAµDψDψexp{iS}(107) in a mathematically precise manner, one usually introduces a gauge–fixing term together with Faddeev–Popov ghosts. E.1.2 New formulation In the present framework, the bosonic gamma matrices Ωµare used to restrict the photon field from the beginning to its two physical components: ZextQED =ZDψDψD(Ω-type) exp iSextQED[ψ, ψ, Ω].(108) The advantages of this form are: 1. The physical degrees of freedom (two transverse components) are selected automatically; 2. Gauge fixing is unnecessary; 3. Ghost fields are unnecessary. F Reference Formulas of the Standard Model and the Present Extension (e–SM) F.1 Reference formulas of the Standard Model (SM) The gauge group is SU(3)c×SU(2)L×U(1)Y. The field strengths and the covariant derivative are Ga µν =∂µGa ν−∂νGa µ+gsfabcGb µGc ν,(109) Wi µν =∂µWi ν−∂νWi µ+g ϵijkWj µWk ν,(110) Bµν =∂µBν−∂νBµ,(111) Dµ=∂µ−igsTaGa µ−ig τi 2Wi µ−ig′Y Bµ.(112) The standard SM Lagrangian then reads LSM =−1 4Ga µνGaµν −1 4Wi µνWiµν −1 4BµνBµν | {z } Lgauge +X ψ ¯ ψ iγµDµψ | {z } Lferm + (DµΦ)†(DµΦ) −V(Φ) | {z } LH +LYuk |{z} Yukawa (113) with the potential and Yukawa sector V(Φ) = µ2Φ†Φ + λ(Φ†Φ)2,LYuk =−¯ QLYdΦdR−¯ QLYu˜ ΦuR−¯ LLYeΦeR+ h.c., (114) 39 where ˜ Φ = iτ2Φ∗,QL, LLare left–handed doublets, and uR, dR, eRare right–handed singlets. After spontaneous symmetry breaking, Aµ Zµ=cos θWsin θW −sin θWcos θWBµ W3 µ, e =gsin θW=g′cos θW. F.2 Notation of the present framework (effective gamma) and replacement rules To embed the metric on the matrix side, we define the two–index gamma and the effective gamma by Γµν(x) := Γµρgρν(x),Γb ν(x) := 3 X µ=0 Γµν(x),{Γb µ(x),Γb ν(x)}= 2 gµν(x)I256.(115) This yields compact expressions with / p(x) := Γb ν(x)pνand D(x) := Γbν(x)∂ν, where the metric gµν is weighted once and only once and the flat limit reproduces the standard 4 ×4 form. In parallel, the bosonic gamma Ω is constructed from the map σ: 0 ↔3,17→ 1,27→2 via Ωµν =1 2(Γµν + Γσ(µ)σ(ν)), and the effective quantity Ωb ν=PµΩµν satisfies {Ωb µ,Ωb ν}= 2gµνI256, as verified in Mathematica. Replacement dictionary SM →e (kinematics / gauge) γµ−→ Γbµ,¯ ψ iγµDµψ−→ ¯ ΨiΓbµ+ ΩbµDµΨ.(116) The U(1) and SU(3) kinetic terms are replaced using a statistical transition function T∈(0,1): −1 4F2−→ −1 4(1 −TγF )F2+TγF T2,−1 4(Ga)2−→ −1 4h(1 −TgF )(Ga)2+TgF (e Ga)2i, (117) where Tµν =∂µBν−∂νBµis the Abeliandualfield in e-QED, and e Ga µν is the fermion–type gluon field–strength in e-QCD. F.3 Extended Standard Model (e–SM) in the present framework Embedding e-QED (Abelian; U(1)×U(1)) and e-QCD (SU(3)) into the full SM, we define LeSM =L(e) gauge +L(e) ferm +LH+LYuk (118) with the following components. (i) Fermion kinetic term (effective gamma + minimal coupling) L(e) ferm =X Ψ∈SM fermions ¯ ΨhiΓbµ+ ΩbµDµ−mΨiΨ,{Ωb µ,Ωb ν}= 2gµν I256.(119) The covariant derivative is Dµ=∂µ−igsTaGa µ−ig τi 2Wi µ−ig′Y Bµ−ig⋆ YYBµ(120) with Bµan additional Abelian field in e-QED (at low energies, Dµ→∂µ+ieAµ+ie′Bµ). 40 (ii) Gauge kinetic terms (dual pair) L(e) gauge =−1 4h(1 −TγF )FµνFµν +TγF Tµν Tµνi | {z } e-QED (Abelian; U(1)×U(1)) (121) −1 4h(1 −TgF )Ga µνGa µν +TgF e Ga µν e Ga µνi | {z } e-QCD (SU(3)) −1 4Wi µνWi µν .(122) In e-QED the longitudinal mode is automatically removed by Ω, so ghosts are unnecessary; in e-QCD the dual kinetic term is added. (iii) Higgs and Yukawa LH= (DµΦ)†(DµΦ) −V(Φ),LYuk = Eq. (113) in the same form,(123) DµΦ = ∂µ−ig τi 2Wi µ−ig′YΦBµ−ig⋆ YYΦBµΦ, YΦ=1 2.(124) F.4 Consistency in limits and replacement summary 1. Flat limit: gµν →ηµν, Γb ν→γν(with unified trace normalization) gives L(e) ferm → Pψ¯ ψ iγµDµψ. The flat–space consistency of the effective gamma follows from the construction of the matrix–valued Dirac operator. 2. Low–energy limit: TγF , TgF →0 implies L(e) gauge → Lgauge; the contribution of Bµ disappears and (118) reduces to (113). 3. Replacement summary (Eqs. (116)–(117)): γµ⇒Γbµ,−1 4F2⇒−1 4(1−TγF )F2+TγF T2,−1 4(Ga)2⇒−1 4(1−TgF )(Ga)2+TgF (e Ga)2. F.5 Coupling to gravity (master action) and ghost–free property Including gravity, Stot =Zd4x√−ghR 16πG +LeSMi(125) so that the variation δgµν yields Gµν = 8πG Θ(eff) µν (sum of usual and dual sources). In the Abelian sector (e-QED) the Ω matrix removes the longitudinal mode; the gauge–fixing Jacobian det □in the path integral degenerates to a field–independent constant, hence Faddeev–Popov ghosts are unnecessary. Remark (practical computation) The Feynman rules in this framework―vertex −igGΓb νand propagator i(/ p−m+iϵ)−1with / p= Γb νpν―are implemented in the same manner as in the flat reference theory. The Clifford relation of the effective gammas and the handling of trace normalizations follow from the construction of the matrix–valued Dirac operator. 41