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 7, 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 3 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 5 2.1 Basic structure of quantum electrodynamics (QED) ............. 6 2.1.1 Gauge–fixing term ........................... 6 2.1.2 Renormalization procedure ....................... 6 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 ............................ 7 2.3 Challenges and limitations of the conventional framework .......... 8 3 Separation of Spin and Statistics in the Dual Description 8 4 Quantum Description of the Metric Tensor and Symmetries 9 4.1 Basic mathematical structure ......................... 9 4.2 Extended Dirac equation ............................ 9 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 ............... 10 4.5 The 256 ×256 bosonic gamma matrices Ω .................. 10 5 Construction of Extended QED and Extended QCD 12 5.1 Construction of extended QED ........................ 12 5.1.1 Extended Lagrangian (e-QED) .................... 12 5.2 Construction of extended QCD ........................ 12 5.2.1 Extended Lagrangian .......................... 12 5.2.2 Physical degrees of freedom of gluons ................. 13 6 Mechanism for the Generation of the Gravitational Field 13 6.1 Generation of gravity in QED ......................... 13 6.2 Generation of gravity in QCD ......................... 14 7 Conclusion 15 8 Acknowledgments 17 A Detailed Computations of Gamma Matrices 18 A.1 Basic 4 ×4 gamma matrices .......................... 18 A.2 Explicit 256 ×256 extension .......................... 19 A.3 Verification of anticommutation relations ................... 19 B On the Physical Interpretation of the 256-Dimensional Extension 19 2 C Proof of Unitarity of the Scattering Matrix 20 C.1 General properties of the S–matrix ...................... 20 C.2 Verification of the Ward–Takahashi identity ................. 21 C.3 Direct proof of unitarity ............................ 21 C.4 Check of gauge invariance ........................... 21 D Proof of Gauge–Fixing–Free Formulation 22 D.0.1 Step 1: Automatic restriction of degrees of freedom ......... 22 D.0.2 Step 2: Preservation of gauge invariance ............... 22 D.0.3 Step 3: Proof of unitarity ....................... 22 E Path Integral for QED Without Gauge Fixing and Ghosts 23 E.1 Theoretical framework ............................. 23 E.1.1 Conventional QED path integral .................... 23 E.1.2 New formulation ............................ 23 F Reference Formulas of the Standard Model and the Present Extension (e–SM) 24 F.1 Reference formulas of the Standard Model (SM) ............... 24 F.2 Notation of the present framework (effective gamma) and replacement rules 24 F.3 Extended Standard Model (e–SM) in the present framework ........ 25 F.4 Consistency in limits and replacement summary ............... 26 F.5 Coupling to gravity (master action) and ghost–free property ........ 26 G Concrete Example: Compton Scattering 26 G.0.1 Calculation in Minkowski spacetime .................. 26 G.1 Metric tensor and extended gamma matrices ................. 27 G.1.1 Application of the extended gammas to Compton scattering . . . . 27 G.1.2 Concrete impact of the metric tensor on Compton scattering . . . . 27 G.1.3 Explicit modifications in the Compton calculation .......... 28 G.1.4 Comparison between Minkowski and curved–spacetime results . . . 28 H Obtaining the Mathematica Codes 29 H.1 Code structure ................................. 29 I Data Availability 30 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 [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. 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. 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. 8 4 Quantum Description of the Metric Tensor and Symmetries In this section we present only the minimal set of definitions and relations required for the present implementation. For derivations, operating rules, and comparisons in scattering calculations, see [17]. We work strictly in four dimensions and build on the two–index gamma matrices Γµν(µ, ν = 0,1,2,3) realized as 256 ×256 constant matrices. By embedding the metric on the matrix side we define the effective gamma Γb νand construct the Dirac operator directly. With this form, general–relativistic effects are tied directly to QFT computation rules (matrix products and traces) without introducing a vierbein or an independent spin connection explicitly. 4.1 Basic mathematical structure 定義 1(Two–index and effective gammas).Let Γµν(µ, ν = 0, . . . , 3) be 256×256 constant matrices satisfying, with the Minkowski metric ηµν = diag(−1,1,1,1), {Γµν,Γρσ}= 2 δν σηµρ I256.(14) Lowering the second index by Γµν := Γµαηανyields {Γµν,Γρσ}= 2 ηµρ ηνσ I256.(15) On a curved spacetime (M, gµν(x)) we define Γµν(x) := Γµρgρν(x),Γb ν(x) := 3 X µ=0 Γµν(x),(16) so that the effective gamma obeys {Γb µ(x),Γb ν(x)}= 2 gµν(x)I256 (17) with the metric weight applied only once as in (16), thereby avoiding any double counting of g2. 4.2 Extended Dirac equation Using the effective gammas, define the Dirac operator by D:= Γbν(x)∂νwith Γbν:= gνρΓb ρ. The generalized Dirac equation then reads iΓbν(x)∂ν−mΨ(x) = 0,(18) which reduces to the standard form in the flat limit gµν →ηµν with Γb ν→γ(b) ν. 9 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. (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). 16 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 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] 17 [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] Becchi, C., Rouet, A., and Stora, R., “Renormalization of Gauge Theories,” Ann. Phys. 98, 287–321 (1975). [14] Kawamura, Y., Gauge Theory from Fundamental Physics (in Japanese), Saiensu–sha, 2017. ASIN: B0785MQP9Q. [15] 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). [16] Sato, H., Groups and Physics (in Japanese), Maruzen Publishing, 2016. ISBN: 978–4621300848. [17] 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    (43a) γ1=i    0001 0010 0100 1000    (43b) γ2=i    000−i 0 0 i0 0−i0 0 i000    (43c) γ3=i    0 0 1 0 0 0 0 −1 −1 0 0 0 0 1 0 0    (43d) These matrices satisfy the anticommutation relations {γµ, γν}= 2gµνI4.(44) 18 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,(45a) σz=1 0 0−1, e =1 0 0 1.(45b) 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.(46) 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),(47a) [σi, e] = 0.(47b) Together with the tensor–product property (A⊗B)(C⊗D) = (AC)⊗(BD),(48) 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.(49) 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: 19 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. 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),(50) where the interaction Hamiltonian is HI=ψigµνΓνDµ+igµν ΩµDνψ. (51) To prove unitarity, we proceed through the following steps. 20 C.2 Verification of the Ward–Takahashi identity First, we confirm invariance under gauge transformations: ψ→e−ieα(x)ψ, (52a) Aµ→Aµ+∂µα(x).(52b) The generating functional is Z[J] = ZDψDψDAµexpiS +iZd4x JµAµ.(53) The Ward–Takahashi identity then reads ∂µ δZ[J] δJµ(x)= 0.(54) This relation is preserved even when the bosonic gamma matrices are used: ∂µhT{jµ(x)Aν(y)}i = 0.(55) C.3 Direct proof of unitarity Expand the scattering operator as S= 1 + iT. (56) The unitarity condition SS†=S†S= 1 implies T−T†=i T†T. (57) To check this, consider 2→2 scattering: hf|T|ii=Mtree +Mloop,(58a) Mloop =Zd4k (2π)4M(k).(58b) With the bosonic gamma matrices, the loop integral can be written as Mloop =Zd4k (2π)4M(k) [ 1 −T(k) ],(59) and in this form one verifies the optical–theorem relation 2 Im M=X nZdΠnMnM∗ n.(60) 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,(61) which follows naturally from the property of the bosonic gammas {Ωµ,Ων}= 2 gµν.(62) 21 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)). 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.(63) From {Ωb µ,Ωb ν}= 2g(Ω) µν I256 it follows immediately that ΩbµAµ2=g(Ω) µν AµAν= (A1)2+ (A2)2,(64) 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 (65) leave the action invariant: δS = 0.(66) (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 (67) is satisfied without introducing gauge fixing or ghost fields. Details of the proof are given in the appendix. 22 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,(68) the photon field Aµpossesses redundancy under the gauge transformation Aµ→Aµ+∂µΛ. Hence, to define Z=ZDAµDψDψexp{iS}(69) 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].(70) 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. 23 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,(71) Wi µν =∂µWi ν−∂νWi µ+g ϵijkWj µWk ν,(72) Bµν =∂µBν−∂νBµ,(73) Dµ=∂µ−igsTaGa µ−ig τi 2Wi µ−ig′Y Bµ.(74) 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 (75) with the potential and Yukawa sector V(Φ) = µ2Φ†Φ + λ(Φ†Φ)2,LYuk =−¯ QLYdΦdR−¯ QLYu˜ ΦuR−¯ LLYeΦeR+ h.c., (76) 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.(77) 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. 24 Replacement dictionary SM →e (kinematics / gauge) γµ−→ Γbµ,¯ ψ iγµDµψ−→ ¯ ΨiΓbµ+ ΩbµDµΨ.(78) 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, (79) 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 (80) 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.(81) The covariant derivative is Dµ=∂µ−igsTaGa µ−ig τi 2Wi µ−ig′Y Bµ−ig⋆ YYBµ(82) with Bµan additional Abelian field in e-QED (at low energies, Dµ→∂µ+ieAµ+ie′Bµ). (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)) (83) −1 4h(1 −TgF )Ga µνGa µν +TgF e Ga µν e Ga µνi | {z } e-QCD (SU(3)) −1 4Wi µνWi µν .(84) 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. (75) in the same form,(85) DµΦ = ∂µ−ig τi 2Wi µ−ig′YΦBµ−ig⋆ YYΦBµΦ, YΦ=1 2.(86) 25