Application of Extended Dirac Equation to Photon–Electron Interactions and Electron–Positron Collision Processes: A Quantum Theoretical Approach Using a 256 × 256 Matrix Representation

We propose a novel theoretical framework for describing photon--electron interactions and electron collision processes in a unified manner within quantum electrodynamics. Specifically, we develop a method to construct the Dirac operator in curved spacetime using only matrix representations rooted in the basis structure of four-dimensional gamma matrix algebra, without introducing vierbeins (tetrads) or independent spin connections. We realize 16 gamma matrices with two indices as $256\times256$ matrices and embed the spacetime metric directly into the matrix elements. This reduces geometric operations such as covariantization, connection-like operations, and basis transformations to matrix products and trace calculations, yielding a unified and transparent computational scheme. The spacetime dimension remains four, and the number ``16'' represents the number of basis elements of four-dimensional gamma matrix algebra ($2^{4}=16$). Based on the extended QED Lagrangian, vertex rules, propagators, spin sums, and traces can be handled uniformly, making it suitable for automation. As validation of this method, we analyzed four fundamental scattering processes in atomic and particle physics: (i) Compton scattering (photon--electron scattering), (ii) muon pair production ($e^+e^-\to\mu^+\mu^-$), (iii) M{\o}ller scattering (electron--electron collision), and (iv) Bhabha scattering (electron--positron collision). In the flat spacetime limit, we confirmed exact reproduction of standard quantum electrodynamics (QED) results including the Klein--Nishina formula. Furthermore, trial calculations using a metric with off-diagonal components show systematic deviations from flat results near scattering angle $\theta\approx90^{\circ}$, suggesting that metric-induced angular dependence could in principle serve as an observable signature. The matrix representation developed in this work enables unified pipeline execution of theoretical calculations for photon interactions and charged particle collision processes, with expected applications to precision calculations in atomic and particle physics.