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

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Abstract

In physics, the semiclassical limit principle asserts that as Planck’s constant ℏ → 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 ε, 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 ε → 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.

Keywords

the semiclassical limit / the noncommutative residue / the perturbations of the Dirac operator / the Kastler–Kalau–Walze-type theorem

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