Dirac Operator in Curved Spacetime via a Clifford-Algebraic 256 × 256 Matrix Representation and Applications to QED Processes

We propose a new method for constructing the Dirac operator in curved spacetime using only a matrix representation grounded in the basis structure of Clifford algebra, without introducing a vierbein or an independent spin connection. Specifically, we introduce sixteen two-index gamma matrices realized as 256×256 matrices and, by embedding the spacetime metric directly into the matrix elements, provide a unified framework that reduces geometric operations (covariantization, connections, basis transformations, etc.) to matrix products and trace operations. Here the spacetime remains four-dimensional, and “16” labels the basis elements of the Clifford (geometric) algebra. Built on an extended quantum electrodynamics (QED) Lagrangian, the approach treats vertex rules, propagators, spin sums, and traces in an integrated manner, offering advantages in transparency and automation of calculations. As validation, we examine Compton scattering, muon pair production, Møller scattering, and Bhabha scattering, showing that off-diagonal metric components impart characteristic angular dependence to differential cross sections, while in the flat-spacetime limit the results agree exactly with standard QED. Furthermore, in trial computations with a toy metric containing off-diagonal components, systematic deviations from the flat-spacetime behavior appear near a scattering angle of about 90°, suggesting that metric-induced angular dependence could serve as an observational signature. Taken together, these results show that a Clifford-algebraic matrix representation provides a practical means of unifying the Dirac operator on curved backgrounds and quantum electrodynamic processes within a single algebraic calculus. Mathematics Subject Classification (2010). Primary 15A66, 83C60; Secondary 81T20, 81V10.