# Dirac Equation in the Presence of a Gravitational Background: Temporal Non Commutative Framework and Vierbein Formalism

## TL;DR

This paper investigates the quantum dynamics of spinor fields in a gravitational background using a non-commutative spacetime framework and vierbein formalism, focusing on first-order curvature effects.

## Contribution

It introduces a non-commutative temporal framework with vierbein formalism to analyze Dirac equations in curved spacetime, including first-order curvature corrections.

## Key findings

- Derived first-order corrections to solutions and Green's functions
- Analyzed effects of gravitational background on non-commutative quantum fields
- Extended DeWitt's approach to non-commutative spacetime

## Abstract

In this paper, we study the non commutative quantum dynamics of the spinorial field in the presence of a gravitational background within DeWitt's approach. By employing the vierbein formalism in the temporal state of non commutative four-dimensional space time, we also analyze the first level correction of the solution and Green's function in terms of the Riemann curvature tensor. The effect of the gravitational background is imposed at least up to the first order of the Riemann curvature tensor.