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 9, 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 2 1.1 Status and challenges of quantum field theory ................ 2 1.2 Our approach .................................. 3 1.3 Organization of the paper ........................... 3 2 Basic structure of conventional quantum field theory 4 2.1 Basic structure of quantum electrodynamics (QED) ............. 4 2.1.1 Gauge–fixing term ........................... 4 2.1.2 Renormalization procedure ....................... 5 2.2 Basic structure of quantum chromodynamics (QCD) ............. 5 2.2.1 Gauge–fixing term ........................... 5 2.2.2 Faddeev–Popov ghost term ....................... 5 2.2.3 BRST symmetry ............................ 6 2.3 Challenges and limitations of the conventional framework .......... 6 3 Separation of Spin and Statistics in the Dual Description 7 4 Quantum Description and Symmetry of the Metric Tensor 9 4.1 Basic Mathematical Structure ......................... 9 4.2 Comparison between the Conventional and Extended Dirac Equations . . . 10 4.3 Detailed structure of the gamma system ................... 10 4.3.1 Construction of the basic operators .................. 10 4.4 Anticommutation relations and metric structure ............... 11 4.5 The 256 ×256 bosonic gamma matrices Ω .................. 11 5 Construction of Extended QED and Extended QCD 13 5.1 Construction of extended QED ........................ 13 5.1.1 Extended Lagrangian (e-QED) .................... 13 5.2 Construction of extended QCD ........................ 13 5.2.1 Extended Lagrangian .......................... 13 5.2.2 Physical degrees of freedom of gluons ................. 14 6 Mechanism for the Generation of the Gravitational Field 14 6.1 Generation of gravity in QED ......................... 14 6.2 Statistical Duality and Atomic-Scale Phase Transition (Conceptual Diagram) 16 6.3 Generation of gravity in QCD ......................... 17 7 Conclusion 18 8 Acknowledgments 19 A Detailed Computations of Gamma Matrices 21 A.1 Basic 4 ×4 gamma matrices .......................... 21 A.2 Explicit 256 ×256 extension .......................... 21 A.3 Verification of anticommutation relations ................... 22 B On the Physical Interpretation of the 256-Dimensional Extension 22 2 C Proof of Unitarity of the Scattering Matrix 23 C.1 General properties of the S–matrix ...................... 23 C.2 Verification of the Ward–Takahashi identity ................. 24 C.3 Direct proof of unitarity ............................ 24 C.4 Check of gauge invariance ........................... 24 D Proof of Gauge–Fixing–Free Formulation 25 D.0.1 Step 1: Automatic restriction of degrees of freedom ......... 25 D.0.2 Step 2: Preservation of gauge invariance ............... 25 D.0.3 Step 3: Proof of unitarity ....................... 25 E Path Integral for QED Without Gauge Fixing and Ghosts 26 E.1 Theoretical framework ............................. 26 E.1.1 Conventional QED path integral .................... 26 E.1.2 New formulation ............................ 26 F Reference Formulas of the Standard Model and the Present Extension (e–SM) 27 F.1 Reference formulas of the Standard Model (SM) ............... 27 F.2 Notation of the present framework (effective gamma) and replacement rules 27 F.3 Extended Standard Model (e–SM) in the present framework ........ 28 F.4 Consistency in limits and replacement summary ............... 29 F.5 Coupling to gravity (master action) and ghost–free property ........ 29 G Concrete Example: Compton Scattering 29 G.0.1 Calculation in Minkowski spacetime .................. 29 G.1 Metric tensor and extended gamma matrices ................. 30 G.1.1 Application of the extended gammas to Compton scattering . . . . 30 G.1.2 Concrete impact of the metric tensor on Compton scattering . . . . 30 G.1.3 Explicit modifications in the Compton calculation .......... 31 G.1.4 Comparison between Minkowski and curved–spacetime results . . . 31 H Obtaining the Mathematica Codes 32 H.1 Code structure ................................. 32 I Data Availability 33 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 3 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. •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. 4 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: •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. 5 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. 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. 6 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. 2.2.3 BRST symmetry The full theory including the gauge–fixing and ghost terms is invariant under the BRST transformations [14]:[15] δBGa µ=Dµca,(10a) δBca=−g 2fabccbcc,(10b) δBca=1 ξ∂µGa µ.(10c) This symmetry guarantees the physical unitarity of the theory. 7 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. (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. 8 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 is the kinetic term of the fermion–type gluon, forming a statistical dual to the usual Ga µνGa µν. Through this duality, as in the QED case, the energy–momentum tensor can feed into the spacetime metric and induce gravitational effects. 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 for the Generation of the Gravitational Field In this section we show, at the level of the energy–momentum tensor, how curvature of the metric gµν (i.e., a gravitational field) emerges naturally from the variation of the action once a statistical duality (a weighted sum of the usual and dual kinetic terms) is introduced in the gauge–field sector. Using the same notation as in the previous section, the total action is Stot =ZM d4x√−ghR 16πG +Lmψ, ¯ ψ;Aµ, Bµ;gµνi,Lm≡ LeQED or LeQCD,(30) where the e-QED Lagrangian (introduced earlier) is ψ[i(Γbµ+Ωbµ)Dµ−m]ψ−1 4FµνFµν − 1 4T(field) µν Tµν (field), with Dµ=∂µ+ieAµ+ie′Bµand T(field) µν =∂µBν−∂νBµ. 6.1 Generation of gravity in QED (i) Variation and the Einstein equation. From the metric variation δStot/δgµν = 0 we obtain Gµν = 8πG Θ(γ) µν ,Θ(γ) µν = Θ(A) µν + Θ(B) µν ,(31) with Θ(A) µν := FµαFνα−1 4gµνFαβFαβ,(32) Θ(B) µν := T(field) µα T(field) ν α−1 4gµνT(field) αβ T(field) αβ.(33) Using the Bianchi identity ∇µGµν ≡0 together with the U(1) field equations, one finds ∇µΘ(γ) µν = 0. 16 (ii) Effective Lagrangian based on duality. Introduce a transition function TγF (E)∈ (0,1) that weights the fermionic component of the photon. In the local–constant (LCM) approximation, LEM, dual(E) = −1 4h(1 −TγF )FµνFµν +TγF T(field) µν T(field) µνi.(34) The corresponding source is Θ(γ) µν (E) = (1 −TγF ) Θ(A) µν +TγF Θ(B) µν .(35) Thus TγF →0 for EEfb (standard QED) and TγF →1 for EEfb (the dual sector dominates). (iii) Electromagnetic invariants and Θ(A) µν Θ(A)µν.Using the Maxwell invariants I1:= FµνFµν = 2(B2−E2), I2:= Fµν ˜ Fµν =−4E·B,˜ Fµν := 1 2εµναβFαβ,(36) one obtains the standard identity Θ(A) µν Θ(A)µν =1 4I2 1+I2 2(37) in particular, for a plane wave (I1=I2= 0), Θ(A) µν Θ(A)µν = 0 (a null stress–energy). (iv) Electrostatic limit (near an atom). For B=0one has I1=−2E2, I2= 0, and Θ(A) 00 =1 2E2,Θ(A) ij =−EiEj+1 2δijE2,Θ(A) µν Θ(A)µν =E4.(38) Similarly, for T(field) µν , Θ(B) µν Θ(B)µν =1 4J2 1+J2 2with J1:= T(field) αβ T(field) αβ and J2:= T(field) αβ ˜ Tαβ (field). (v) Newtonian limit and a local Poisson equation. With g00 =−1−2Φ and weak spatial curvature, G00 ≃2∇2Φ gives ∇2Φ = 4πG ρeff, ρeff = Θ(γ) 00 = (1 −TγF )1 2(E2+B2) + TγF 1 2(C2+D2),(39) where T(field) 0i=Ciand T(field) ij =εijkDk. As TγF (E) increases in high–density regions such as inside atoms, ρeff is enhanced and the local potential Φ deepens, providing an induced–gravity contribution from the dual sector. Remark (effective two–dimensionality from Ω). From the previously noted properties of the bosonic gamma Ω, quantities involving Ω effectively receive contributions only from the (1,2) components of gµν (only the central 2 ×2 block is nonzero). In the Minkowski metric one has (PgP⊤)µν = diag(0,1,1,0), so the longitudinal/timelike directions (0,3) drop out automatically. 17 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: Schematic illustration of fermion–boson duality (concept of atomicscale phase transition). Left: fermionic dominance (repulsive tendency); right: bosonic dominance (attractive tendency); center: phase transition inside an atom. The lower panel corresponds to the table of four “types” of particles, where the notation of mass represents an effective mass within a medium. This figure visualizes the transition function TγF (E) introduced in Sec. 3and the interpretation of the gravitational source term derived from the Einstein equations (31)–(39). 6.2 Statistical Duality and Atomic-Scale Phase Transition (Conceptual Diagram) Explanation of Fig. 2(Phenomenological Summary) Working Hypothesis: Atomic-Scale Phase Transition and Enhancement of the Gravitational Source. When the fermion–boson duality is introduced into the Standard Model, the transition function TγF (E) (and similarly TXfor other fields) modifies the weighted sum of the normal and dual terms in the Lagrangian density. Through this mechanism, the right-hand side (source term) of the Einstein equation, Gµν = 8πG Θ(γ) µν , becomes enhanced. In regions such as atomic nuclei or strongly bound domains, the bosonic (or dual) components increase, causing the local potential Φ to deepen according to Eq. (39). Interaction Tendencies (F–F Repulsion / B–B Attraction). When fermionic components dominate, the repulsive tendency arises from degeneracy pressure and the exclusion principle. When bosonic components dominate, coherent binding (attractive tendency) becomes effective due to condensation phenomena. These are not universal laws but rather effective statistical tendencies in many-body systems. The manifestation of gravity is ultimately evaluated through Θµν (see Eq. (31)). On the Statement Matter is a Superconductor/Superfluid. In this paper, it is treated as a working hypothesis thatwithin atoms or in high-density regions, local superconducting or superfluid-like components may dominate. This does not imply that the bare masses or statistics of electrons and photons in vacuum change permanently. 18 Table 1: Characteristics of the four components corresponding to the lower panel (schematic reconstruction). Particle type Mass Description Fermionic electron Present The ordinary, experimentally observed electron. Fermionic photon Present Mode possessing an effective mass within a medium, corresponding to the Meissner effect. Bosonic electron Absent Collective excitation in a condensed phase with effective mass ≈0 (e.g., phase mode). Bosonic photon Absent The ordinary, observable photon. Note: Heremass denotes an effective quantity within a medium (or condensed phase), not the bare vacuum mass. The terms massive and massless in the figure indicate effective mass values within media, distinct from the intrinsic bare mass in vacuum. Summary Using the QED-side equations (31)–(39), this section demonstrates that the duality modifies the right-hand side of the Einstein equation through the weighting of the energy–momentum tensor, and that the increase of TγF (E) can locally enhance the effective gravitational source. Figure 2visualizes this physical intuition, illustrating the effective tendencies of F–F repulsion and B–B attraction, and the working hypothesis of atomicscale phase transition. 6.3 Generation of gravity in QCD (i) Dual structure on the e-QCD side. Write the e-QCD kinetic sector as LYM, dual(E) = −1 4h(1 −TgF )Ga µνGa µν +TgF e Ga µν e Ga µνi, Ga µν =∂µGa ν−∂νGa µ+gsfabcGb µGc ν, (40) where e Ga µν denotes the fermion–type gluon field–strength. The associated stress–energy tensors are Θ(G) µν =Ga µαGaνα−1 4gµν Ga αβGa αβ,(41) e Θ(G) µν =e Ga µα e Gaνα−1 4gµν e Ga αβ e Ga αβ.(42) Hence the gravitational source becomes Gµν = 8πGh(1 −TgF ) Θ(G) µν +TgF e Θ(G) µν i.(43) When TgF (E) is large in high–density regions such as nucleon interiors, the contribution of e Θ(G) µν becomes relatively dominant, and one expects stronger local gravitational effects than in QED. 19 (ii) Appearance in the Newtonian limit. In the center–of–mass approximation one has ∇2Φ = 4πG ρ(QCD) eff , with ρ(QCD) eff = (1 −TgF )1 2Ea G 2+Ba G 2+TgF 1 2e Ea2+e Ba2,(44) where Ga 0i=Ea G i and Ga ij =εijkBa G k (and similarly for e G). A large TgF deepens Φ and yields an enhancement of gravitational effects on nucleon scales. Summary For QED, Eqs. (31)–(39), and for QCD, Eqs. (43)–(44), show that the usual terms FµνFµν /Ga µνGa µν and the dual terms T(field) µν T(field) µν /e Ga µν e Ga µν are superposed through the transition functions TγF (E) and TgF (E), forming the effective gravitational sources Θ(γ) µν and Θ(QCD) µν . Inside atoms and nucleons the dual weights grow and strengthen the local potential Φ. 7 Conclusion In this work we introduced a framework based on two–index gamma matrices that embed the metric on the matrix side and on the effective gamma (column sum) Γµν(x) := Γµρgρν(x),Γb ν(x) := 3 X µ=0 Γµν(x), with which {Γb µ(x),Γb ν(x)}= 2 gµν(x)I256. Based on these relations we defined a matrix–valued Dirac operator D(x) = Γbν(x)∂ν and provided a framework in which quantum electromagnetic processes on curved spacetimes are handled without explicitly introducing vierbeins or an independent spin connection, using only matrix operations (products and traces) in the Clifford algebra. In the flat limit the formulation agrees exactly with the standard 4 ×4 representation. This minimal setup furnishes a single–rule computational basis for vertices, propagators, spin sums, and traces; we validated it in QED processes such as Compton scattering. (1) e-QED / e-QCD and the master action. On a curved background (M, gµν) we formulated extended QED as LeQED =ψhiΓbµ+ ΩbµDµ−miψ−1 4FµνFµν −1 4T(field) µν Tµν (field), Stot =Zd4x√−ghR 16πG +LeQEDi, from which the variation δgµν yields Gµν = 8πGΘ(A) µν + Θ(B) µν (A: ordinary photon, B: fermion–type photon). In the U(1)×U(1) sector the longitudinal modes are automatically removed by the Ω matrices, the gauge–fixing Jacobian in the path integral degenerates to a field–independent constant, and hence Faddeev–Popov ghosts are unnecessary. Extending to SU(3) (e-QCD), the dual pair Ga µνGa µν and e Ga µν e Ga µν contributes to the gravitational source. 20 (2) Explicit gravitational sources through duality. With transition functions TγB(E), TγF (E) (TγB +TγF = 1), the effective electromagnetic Lagrangian and the gravitational source read Leff EM =−1 4hTγBFµνFµν +TγF T(field) µν Tµν (field)i,Θ(γ) µν =TγBΘ(A) µν +TγF Θ(B) µν . In the Newtonian limit this gives ∇2Φ = 4πG [TγBρA+TγF ρB+ρ(ψ)],so that in high–density regions an increase of TγF enhances ρBand thus local gravity. On the QCD side, introducing TgB, TgF one can likewise write Θ(gluon) µν =TgBΘ(G) µν +TgF e Θ(G) µν ,which points to astrengthening of the gravitational source at the nucleon scale. (3) Computational checks and consistency in the flat limit. Using the Feynman rules in this framework (vertex −ie Γb ν, propagator i(/ p−m+iϵ)−1with / p:= Γb νpν), we compared (A) the standard 4 ×4 evaluation, (B) a flat 256 ×256 realization, and (C) a 256 ×256 realization with the metric embedded, for benchmark QED processes including Compton scattering. After aligning normalizations we find exact agreement between (A) and (B), while (C) exhibits significant angular dependence in toy models with non–diagonal metrics. This exact agreement in the flat limit underpins the consistency of the present approach. (4) Physical implications. (i) Gravitational consequences of statistical duality: in high–density/high–energy regimes the dual sector can dominate, so that Θ(B) µν (QED) or e Θ(G) µν (QCD) can amplify the effective gravitational source. (ii) Practical simplification: with unified matrix rules, curvature effects reduce to metric substitution plus trace calculations. (iii) Observability: while effects at atomic scales are tiny, collective manifestations may arise at the nucleon scale or in high–density matter. (5) Limitations and future work. Our numerical examples used local–constant approximations (LCM) in toy models. A systematic treatment on fully position–dependent backgrounds (including connections), loop calculations and re–regularization, first–order corrections in weak gravity and gravitational–wave perturbations, and quantitative tests in the strong–coupling QCD regime are required. We plan to implement these sequentially, aiming to connect to phenomenology while retaining the advantages of the matrix–embedded approach (automation and portability). Summary. The 256 ×256 matrix representation based on two–index and effective gammas provides (1) a concise means to embed curvature on the matrix side while remaining fully four–dimensional; (2) a constructive realization of Gµν = 8πG Θ(eff) µν through the dual kinetic pair in U(1)×U(1) and SU(3); and (3) exact consistency with standard QED/QCD in the flat limit together with observable corrections on curved backgrounds. The framework functions as a gauge–fixing–free, ghostless and automation–friendly computational basis, offering a unified perspective on the emergence of gravity via statistical duality. 8 Acknowledgments The present study benefited decisively from e–mail discussions with a full professor and an associate professor specializing in particle physics, which inspired the core idea of fermion–boson duality developed in this paper. I am deeply grateful for numerous insights 21 into the mathematical structure and physical meaning of the theory gained through those conversations. I also express my sincere thanks for editorial assistance provided by the generative AIs ChatGPT and Claude. References [1] Griffiths, D. J., Introduction to Elementary Particles, 2nd Rev. Ed., Wiley–VCH, 2008. ISBN: 978–3527406012. [2] Schwinger, J., “On Quantum Electrodynamics and the Magnetic Moment of the Electron,” Phys. Rev. 73, 416–417 (1948). [3] Kinoshita, T., “The Anomalous Magnetic Moment of the Electron and the Muon,” Rev. Mod. Phys. 81, 865–934 (2009). [4] Gross, D. J., and Wilczek, F., “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., and Gross, D. J., “Asymptotic Freedom in the Parton Model,” 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., and Mills, R. L., “Conservation of Isotopic Spin and Isotopic Gauge Invariance,” Phys. Rev. 96, 191–195 (1954). [12] Faddeev, L. D., and Popov, V. N., “Feynman Diagrams for the Yang–Mills Field,” Phys. Lett. B 25, 29–30 (1967). [13] H. Maruyama, “Dirac Operator in Curved Spacetime via a Clifford-Algebraic 256×256 Matrix Representation and Applications to QED Processes,” Frontiers in Physics (2025). DOI: 10.3389/fphy.2025.1618853. [14] Becchi, C., Rouet, A., and Stora, R., “Renormalization of Gauge Theories,” Ann. Phys. 98, 287–321 (1975). [15] Kawamura, Y., Gauge Theory from Fundamental Physics (in Japanese), Saiensu–sha, 2017. ASIN: B0785MQP9Q. 22 [16] Klein, O., and Nishina, Y., “On the Scattering of Radiation by Free Electrons According to Dirac’s New Relativistic Quantum Mechanics,” Z. Phys. 52, 853–868 (1929). [17] Sato, H., Groups and Physics (in Japanese), Maruzen Publishing, 2016. ISBN: 978–4621300848. [18] Maruyama, H., “Dirac Operator in Curved Spacetime via a Clifford–Algebraic 256×256 Matrix Representation and Applications to QED Processes,” Frontiers in Physics, 2025. 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    (45a) γ1=i    0001 0010 0100 1000    (45b) γ2=i    000−i 0 0 i0 0−i0 0 i000    (45c) γ3=i    0 0 1 0 0 0 0 −1 −1 0 0 0 0 1 0 0    (45d) These matrices satisfy the anticommutation relations {γµ, γν}= 2gµνI4.(46) 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,(47a) σz=1 0 0−1, e =1 0 0 1.(47b) 23 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.(48) 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),(49a) [σi, e] = 0.(49b) Together with the tensor–product property (A⊗B)(C⊗D) = (AC)⊗(BD),(50) 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.(51) 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. 24 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. 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),(52) where the interaction Hamiltonian is HI=ψigµνΓνDµ+igµν ΩµDνψ. (53) To prove unitarity, we proceed through the following steps. 25 G.1 Metric tensor and extended gamma matrices In what follows we employ the two–index gammas Γµνand the effective gamma Γb ν,2 where the two–index basis can be explicitly realized (16 elements) as an eightfold tensor product of Pauli matrices (see Sec. 5 of this paper), and satisfies the Clifford relation {Γµν,Γρσ}= 2 δνσηµρ I256 . On curved spacetimes, the metric is embedded once on the matrix side so that {Γb µ(x),Γb ν(x)}= 2 gµν(x)I256 holds (avoiding any double counting of the metric). The extended Dirac equation then reads iΓbν(x)∂ν−mΨ(x) = 0,Γbν:= gνρΓb ρ and reduces to the standard form in the flat limit. G.1.1 Application of the extended gammas to Compton scattering Within the present framework, QED calculations on curved backgrounds proceed in the same way as in the flat case by simply replacing the conventional γµwith γµ−→ Γbµ(x). The key changes are: 1. Instead of the four 4 ×4 matrices γµ, one uses the effective gammas Γb νconstructed from the sixteen 256 ×256 two–index basis elements Γµν. 2. Replace the flat metric ηµν with a general metric gµν(x). 3. Use the Clifford relation {Γb µ,Γb ν}= 2gµνI256 (vertex −ie Γb µ, slash p/(x) = Γb ν(x)pν). G.1.2 Concrete impact of the metric tensor on Compton scattering In general, gµν(x) =     g00 g01 g02 g03 g10 g11 g12 g13 g20 g21 g22 g23 g30 g31 g32 g33    ,(92) and in Minkowski spacetime gµν =ηµν = diag(−1,1,1,1) .As a curved–spacetime toy example, we consider a case with small off–diagonal components: gµν =    −0.999 0.001 0.001 0.001 0.001 1 0.001 0.001 0.001 0.001 1 0.001 0.001 0.001 0.001 1    ,(93) which models a slight spatial distortion. Consistency under general coordinate transformations is preserved by regarding Γb νas a covariant vector. 2Definitions: Γµν (x) := Γµ ρgρ ν(x), Γb ν(x) := P3 µ=0 Γµν (x). Then {Γb µ,Γb ν}= 2gµν I256. 32 G.1.3 Explicit modifications in the Compton calculation The computation with extended gammas and a non–flat metric proceeds as follows: 1. Matrices and metric setup: prepare the sixteen two–index basis elements Γµν (via eightfold Pauli tensor products) and specify gµν (x). Γµν(x) = Γµρgρν(x),Γb ν(x) = 3 X µ=0 Γµν(x). 2. Spinorial quantities: the slashed momentum / p(x) = Γb ν(x)pνand the mass m. 3. Trace calculations: f(s, u) = 1 4(s−m2)2Trh(/ p′+m) Γbµ(/ p+/ κ+m) Γbν(/ p+m) Γb ν(/ p+/ κ+m) Γb µi, (94) g(s, u) = 1 4(s−m2)(u−m2)Trh(/ p′+m) Γbµ(/ p+/ κ+m) Γbν(/ p+m) Γb µ(/ p−/ κ′+m) Γb νi. (95) 4. Differential cross section: dσ =dt πe4 (s−m2)2[f(s, u) + g(s, u) + f(u, s) + g(u, s) ].(96) Note. Because the 256×256 representation has a different trace normalization (tr256(I) = 256), the flat–space result agrees exactly with the standard 4×4 result once normalizations are aligned. G.1.4 Comparison between Minkowski and curved–spacetime results We compare the following three settings: A. gµν =ηµν with the conventional 4 ×4γµcalculation. B. gµν =ηµν using Γb νconstructed from the sixteen 256 ×256 basis elements Γµν(flat case). C. gµν 6=ηµν using Γb ν(x) constructed from the sixteen 256×256 basis elements (curved case). Cases A and B are mathematically equivalent and reduce to the standard Klein–Nishina formula: dσ =r2 e 2ω′ ω2ω ω′+ω′ ω−sin2θdθ′.(97) In contrast, in Case C the off–diagonal metric components modify the cross section. As a concrete example (γ= 0.173): dσ = 1.90285 ×10−24 × 1.07406 ×1025 (1.173 −0.173 cos θ)2−2.37098 ×1023 (−1.173 + 0.173 cos θ)3 +1.84352 ×1025 (−1.173 + 0.173 cos θ)+ 7.98326 ×1024!.(98) 33 As shown in Fig. 3, this result clearly deviates from the conventional Klein–Nishina formula and quantifies curvature effects through the specific metric values used. Figure 3: Comparison of the conventional calculation with the present one in which explicit metric components encode spacetime curvature and its impact on the Compton process. H Obtaining the Mathematica Codes The Mathematica codes and example outputs (PDFs) used in this study are publicly available as an auxiliary archive prepared by the author on Zenodo: https://zenodo.org/records/17546599 This record contains minimal runnable notebooks for foundational checks of the theory (Γ/Ω matrices) and representative tree–level processes in e–QED. An overview of the contents is as follows. H.1 Code structure •QED processes (e–QED; tree level) –Compton scattering (eγ →eγ) –Bhabha scattering (e+e−→e+e−) –Møller scattering (e−e−→e−e−) –Muon–pair production (e+e−→µ+µ−) •Theoretical foundations (verification of matrix properties) –Properties of bosonic ω/Ω gamma matrices (compact tables for squares and anticommutators) 34 –Definitions of the four–dimensional and 256 ×256 two–index gammas Γµν and verification of the anticommutation relations (mixed anticommutators) •Minimal workflow for scattering calculations –Kinematic setup (Mandelstam variables, etc.) –Construction of scattering amplitudes (vertices and spin sums) –Derivation of cross sections (angular distributions and totals) –Comparison with the Klein–Nishina formula (Compton scattering) –Checks of high–energy scaling (e.g., s→ ∞) These codes not only demonstrate concrete applications of the present framework but also enable a head–to–head comparison with known results of standard QED in the flat limit. In particular, the construction of the bosonic gamma matrices and the comparison for Compton scattering are crucial for validating the core predictions of the theory. I Data Availability Supplementary data and the Mathematica notebooks associated with this paper are published and versioned in the following Zenodo record: •Zenodo (complete auxiliary package; including PDF outputs and corresponding notebooks): https://zenodo.org/records/17546599 When citing, please include the DOI/URL of the above record (and, where appropriate, the specific version). 35