In physics, the semiclassical limit principle asserts that as Planck's constant $ \hbar \rightarrow 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 $ \varepsilon $, 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 $ \varepsilon \rightarrow 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.
Citation: Tong Wu, Yong Wang. The semiclassical limit of the Kastler–Kalau–Walze-type theorem[J]. Electronic Research Archive, 2025, 33(4): 2452-2474. doi: 10.3934/era.2025109
In physics, the semiclassical limit principle asserts that as Planck's constant $ \hbar \rightarrow 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 $ \varepsilon $, 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 $ \varepsilon \rightarrow 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.
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Tong Wu, Yong Wang. The semiclassical limit of the Kastler–Kalau–Walze-type theorem[J]. Electronic Research Archive, 2025, 33(4): 2452-2474. doi: 10.3934/era.2025109
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