The problem of Chern-Simons-like term induction via quantum corrections in four-dimensions is investigated in two different cases. In the first case, we consider two distinct approaches to deal with the exact fermion propagator of the extended QED theory up to the first order in the $b$-coefficient. We find different results for distinct approaches in the same regularization scheme. In the second case, we show that when we use a modified derivative expansion method and another regularization scheme, we obtain a result that exactly coincides with one of the results obtained in the former case. This seems to imply an ambiguity absence as one treats the fermion propagator and the self-energy tensor properly.

In this work, I investigate the quantum dynamics of a spin-1/2 particle propagating in the spacetime of a static, spherically symmetric Einstein-Gauss-Bonnet (EGB) black hole within the framework of relativistic quantum mechanics. Starting from the Dirac equation in curved spacetime-formulated using the tetrad formalism and the associated spin connection we construct the corresponding Dirac Hamiltonian in EGB geometry. Next, I employ Heisenberg's equations of motion to derive explicit operator expressions for the particle's velocity and force, providing a fully quantum description of fermionic motion in a higher-curvature gravitational background. It is observed that the spacetime geometry modifies the Dirac dynamics through the EGB metric function, leading to corrections in the velocity and force operators that depend explicitly on the Gauss-Bonnet coupling parameter, $\xi$. In the weak-field limit, the expectation values of these operators satisfy Ehrenfest's theorem, demonstrating that the corresponding classical Einstein-Gauss-Bonnet dynamics emerges naturally as the semiclassical limit of the underlying quantum theory. In particular, the effective radial force includes higher-curvature contributions that become increasingly significant in the strong-gravity regime, while continuously reducing to the Schwarzschild result as the Gauss-Bonnet parameter approaches zero. These results establish a direct connection between relativistic quantum dynamics and modified gravity, providing an operator-based framework for investigating fermionic motion in Einstein-Gauss-Bonnet spacetimes.