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arXiv:2207.01080 · General Relativity and Gravitation

Spacetime entanglement entropy: covariance and discreteness

Abhishek Mathur, Sumati Surya, X Nomaan

Published Jul 1, 2022• 4 citations
arXivHTMLPDF
DOI
10.1007/s10714-022-02948-x

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Entanglement entropy (EE) is a pivotal quantity in quantum field theory (QFT) that captures the nonlocal correlations between subsystems, offering deep insights into the quantum structure of spacetime, field dynamics, and phase transitions. This work derives EE in QFT using the covariance matrix formalism, establishing its foundations through the von Neumann entropy of reduced density matrices. We analytically compute EE for coupled harmonic oscillators as a prototypical model, extending the results to quantum fields via perturbative expansions. In the context of Maxwell QFT, we analyze entropy corrections induced by external perturbations, revealing their fractal dynamics through the emergence of Julia sets. The study further incorporates noncommutative geometry, where the deformation of spacetime modifies the covariance matrix, field strength tensors, and modular indices. Numerical simulations validate the derived scaling laws, entropy corrections, and noncommutative effects. These findings provide a mathematically rigorous framework to explore the interplay between EE, field theory, and geometric structures in high-energy physics.
Abstract In the context of the AdS/CFT correspondence, it is often convenient to regulate infinite quantities in asymptotically anti-de Sitter spacetimes by introducing a sharp cutoff at some finite, large value of a particular radial coordinate. This procedure is a priori coordinate dependent, and may not be well-motivated in full, covariant general relativity; however, the fact that physically meaningful quantities such as the entanglement entropy can be obtained by such a regulation procedure suggests some underlying covariance. In this paper, we provide a careful treatment of the radial cutoff procedure for computing holographic entanglement entropy in asymptotically anti-de Sitter spacetimes. We prove two results that are frequently assumed in the literature, but that have not been carefully addressed: (i) that the choice of a “globally minimal surface” among several extremal candidates is independent of the choice of regulator, and (ii) that finite CFT quantities such as the mutual information which involve “divergence-cancelling” sums of entanglement entropies are well-defined under the usual prescription for computing covariant holographic entanglement entropy. Our results imply that the “globally minimal surface” prescription for computing the holographic entanglement entropy is well-posed from the perspective of general relativity, and thus support the widely-held belief that this is the correct prescription for identifying the entanglement wedge of a boundary subregion in AdS/CFT. We also comment on the geometric source of state-dependent divergences in the holographic entanglement entropy, and identify precisely the regime of validity of the “vacuum subtraction” protocol for regulating infinite entanglement entropies in arbitrary states by comparing them to the entanglement entropies of identical regions in the vacuum. Our proofs make use of novel techniques for the covariant analysis of extremal surfaces, which are explained in detail and may find use more broadly in the study of holographic entanglement entropy.