256×256 Dirac Equation in Curved Spacetime: A New Method for Quantum Electrodynamics Calculations without Vierbein Fields

This research proposes a new method for directly handling the Dirac equation in curved spacetime without introducing vierbein fields or spin connections. Conventionally, the Dirac equation in general relativistic environments connects internal space and external space through vierbein fields. However, this research develops a technique that introduces sixteen gamma matrices represented by 256×256 matrices, incorporating curved spacetime metrics directly into matrix elements. This approach is based on an extended QED Lagrangian and offers the advantage of processing conventional complex geometric operations through matrix multiplication and trace operations. Calculations of quantum electrodynamic processes such as Compton scattering, Møller scattering, and Bhabha scattering have shown that non-diagonal components of the metric have characteristic effects on scattering cross-sections. In particular, while the results exactly match conventional results in the flat spacetime limit, observable differences occur at specific scattering angles in curved spacetime. The paper also discusses the possibility of this approach as a mechanism for suppressing divergences in high-energy regions and methods for natural integration into the Standard Model.