On Geometric Entropy

TL;DR

This work defines geometric entropy for quantum fields via tracing over degrees of freedom across a boundary, showing it equals the first quantum correction to Bekenstein-Hawking entropy in the appropriate limit and diverges without a UV cutoff. It employs the replica trick and conical deficit geometries to relate $S_{geom}$ to derivatives of the partition function on deficit-angle manifolds, and it presents explicit calculations in 1D and via heat kernels to reveal boundary-dominated, UV-divergent behavior. The results connect horizon correlations in flat space to black hole thermodynamics and provide a framework for finite-state, shape, and topological refinements, with implications for string theory and orbifold techniques. Overall, the paper demonstrates that geometric (entanglement) entropy encodes the leading quantum corrections to gravitational entropy, while highlighting fundamental divergence issues that challenge renormalization in local quantum field theories.

Abstract

We show that a geometrical notion of entropy, definable in flat space, governs the first quantum correction to the Bekenstein-Hawking black hole entropy. We describe two methods for calculating this entropy -- a straightforward Hamiltonian approach, and a less direct but more powerful Euclidean (heat kernel) method. The entropy diverges in quantum field theory in the absence of an ultraviolet cutoff. Various related finite quantities can be extracted with further work. We briefly discuss the corresponding question in string theory.