The semiclassical limit of the Kastler–Kalau–Walze-type theorem

In physics, the semiclassical limit principle asserts that as Planck's constant $\hslash \to 0$, quantum states reduce to classical configurations. We extend this framework to the noncommutative residue by applying the semiclassical limit to the spectral geometry. By introducing the coefficient $\epsilon$, we establish a proof of the Kastler–Kalau–Walze-type theorem for the perturbations of the Dirac operator on four-dimensional compact manifolds with (without) boundary. As $\epsilon \to 0$, we demonstrate the emergence of a semiclassical limit, thereby providing the classical formulation of the theorem. This result elucidates the interplay between quantum corrections and classical geometric invariants in the presence of boundary conditions.