Spectral Entropy Functional on Compact Three-Manifolds: A Unified Limiting Framework for Geometric Entropy, Statistical Entropy, and von Neumann Entropy
The paper constructs a spectral entropy functional unifying geometric, statistical, and von Neumann entropy in three-manifolds, suggesting a new mathematical framework.
Key Points
The research aims to unify three distinct types of entropy—geometric, statistical, and von Neumann—using a common spectral entropy functional on three-manifolds.
Constructed a global spectral entropy functional on three-dimensional Riemannian manifolds using the eigenspectrum of the Laplace-Beltrami operator.
Proved asymptotic equivalences of the spectral entropy to geometric, Boltzmann, and von Neumann entropies under different limits.
Defined a topological compactness parameter and explored its relationship with the spectral entropy functional.
Established strict asymptotic equivalence of spectral entropy with Riemannian heat-kernel entropy in the UV limit.
Demonstrated that spectral entropy corresponds to Boltzmann counting entropy in the IR limit.
Reproduced the Bekenstein–Hawking entropy expression for static black-hole metrics, confirming the framework's applicability.