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arXiv:gr-qc/9602011 · Modern Physics Letters A

Quantum Decoherence in a Four-Dimensional Black Hole Background

John R. Ellis, Nick E. Mavromatos, Elizabeth Winstanley, D.V. Nanopoulos

Published Feb 10, 1997• 32 citations
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Abstract

We display a logarithmic divergence in the density matrix of a scalar field in the presence of an Einstein–Yang–Mills black hole in four dimensions. This divergence is related to a previously-found logarithmic divergence in the entropy of the scalar field, which cannot be absorbed into a renormalization of the Hawking–Bekenstein entropy of the black hole. Motivated by the fact that the cutoff in this divergence varies as the latter decays, by an analysis of black holes in two-dimensional string models and by studies of D-brane dynamics in higher dimensions, we propose that the renormalization scale variable be identified with time. In this case, the logarithmic divergence we find induces a non-commutator term [Formula: see text] in the quantum Liouville equation for the time evolution of the density matrix ρ of the scalar field, leading to quantum decoherence. The order of magnitude of [Formula: see text] is μ2/M P , where μ is the mass of the scalar particle.

DOI
10.1142/s0217732397000248

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We display a logarithmic divergence in the density matrix of a scalar field in the presence of an Einstein–Yang–Mills black hole in four dimensions. This divergence is related to a previously-found logarithmic divergence in the entropy of the scalar field, which cannot be absorbed into a renormalization of the Hawking–Bekenstein entropy of the black hole. Motivated by the fact that the cutoff in this divergence varies as the latter decays, by an analysis of black holes in two-dimensional string models and by studies of D-brane dynamics in higher dimensions, we propose that the renormalization scale variable be identified with time. In this case, the logarithmic divergence we find induces a non-commutator term [Formula: see text] in the quantum Liouville equation for the time evolution of the density matrix ρ of the scalar field, leading to quantum decoherence. The order of magnitude of [Formula: see text] is μ2/M P , where μ is the mass of the scalar particle.
The entanglement entropy correlates two quantum subsystems which are part of a larger system. A logarithmic divergence term present in the entanglement entropy is universal in nature and directly proportional to the conformal anomaly. We study this logarithmic divergence term of entropy for massive scalar field in [Formula: see text] dimensions by applying numerical techniques to entanglement entropy approach. This (2+1)-dimensional massive theory can be obtained from the (3+1)-dimensional massless scalar field via dimensional reduction. We also calculated mass corrections to entanglement entropy for scalar field. Finally, we observe that the area law contribution to the entanglement entropy is not affected by this mass term and the universal quantities depend upon the basic properties of the system.
Black holes in general relativity carry an entropy whose value is given by the Bekenstein-Hawking formula, but whose statistical origin remains obscure. Such horizons also possess an entanglement entropy, which has a clear statistical meaning but no a priori relation to the dynamics of gravity. For free minimally-coupled scalar and spinor fields, these two quantities are intimately related: the entanglement entropy is the one-loop correction to the black hole entropy due to renormalization of Newton's constant. For gauge fields, the entanglement entropy and the one-loop correction to the black hole entropy differ. This dissertation addresses two issues concerning the entanglement entropy of gauge fields, and its relation black hole entropy. First, for abelian gauge fields Kabat identified a negative divergent contribution to the black hole entropy that is not part of the entanglement entropy, known as a ``contact term''. We show that the contact term can be attributed to an ambiguous expression for the gauge field's contribution to the Wald entropy. Moreover, in two-dimensional de Sitter space, the contact term arises from an incorrect treatment of zero modes and is therefore unphysical. In a manifestly gauge-invariant reduced phase space quantization of two-dimensional gauge theory, the gauge field contribution to the entropy is positive, finite, and equal to the entanglement entropy. This suggests that the contact term in more than two dimensions may also be unphysical. Second, we consider lattice gauge theory and point out that the Hilbert space corresponding to a region of space includes edge states that transform nontrivially under gauge transformations. By decomposing these edge states in irreducible representations of the gauge group, the entanglement entropy of an arbitrary state is shown to be a sum of a bulk entropy and a boundary entropy associated to the edge states. This entropy formula agrees with the two-dimensional results from the reduced phase space quantization. These results are applied to several examples, including the ground state in the strong coupling expansion of Kogut and Susskind, and the entropy of the edge states is found to be the dominant contribution to the entanglement entropy.
Using the AdS/CFT correspondence we derive a formula for the entanglement entropy of the anti-de Sitter black hole in two spacetime dimensions. The leading term in the large black hole mass expansion of our formula reproduces exactly the Bekenstein–Hawking entropy SBH, whereas the subleading term behaves as lnSBH. This subleading term has the universal form typical for the entanglement entropy of physical systems described by effective conformal fields theories (e.g. one-dimensional statistical models at the critical point). The well-known form of the entanglement entropy for a two-dimensional conformal field theory is obtained as analytic continuation of our result and is related with the entanglement entropy of a black hole with negative mass.
We explore the method of entanglement entropy applied to 2d black holes. We introduce a solvable model of a real scalar field with finite volume and lattice spacing in terms of NNN coupled mechanical oscillators and compute its entanglement entropy in many cases. The large NNN limit of this scheme, with finite lattice spacing, should reproduce the Bekenstein-Hawking formula for black hole entropy.
Within the general framework of Liouville string theory, we construct a model for quantum D-brane fluctuations in the space–time background through which light closed-string states propagate. The model is based on monopole and vortex defects on the worldsheet, which have been discussed previously in a treatment of (1+1)-dimensional black hole fluctuations in the space–time background, and makes use of a T-duality transformation to relate formulations with Neumann and Dirichlet boundary conditions. In accordance with previous general arguments, we derive an open quantum-mechanical description of this D-brane foam which embodies momentum and energy conservation and small mean energy fluctuations. Quantum decoherence effects appear at a rate consistent with previous estimates.
We study a proposed statistical explanation for the Bekenstein-Hawking entropy of a black hole in which entropy arises quantum-mechanically as a result of entanglement. Arguments for the identification of black hole entropy with entanglement entropy are reviewed in the framework of quantum field theory, emphasizing the role of renormalization and the need for a physical short-distance cutoff. \n \nOur main novel contribution is a calculation of entanglement entropy in loop quantum gravity. The kinematical Hilbert space and spin network states are introduced, and the entanglement entropy of these states is calculated using methods from quantum information theory. The entanglement entropy is compared with the density of states previously computed for isolated horizons in loop quantum gravity, and the two are found to agree up to a topological term. \n \nWe investigate a conjecture due to Sorkin that the entanglement entropy must be a monotonically increasing function of time under the assumption of causality. For a system described by a finite-dimensional Hilbert space, the conjecture is found to be trivial, and for a system described by an infinite-dimensional Hilbert space a counterexample is provided. \n \nFor quantum states with Euclidean symmetry, the area scaling of the entanglement entropy is shown to be equivalent to the strong additivity condition on the entropy. The strong additivity condition is naturally interpreted in information-theoretic terms as a continuous analog of the Markov property for a classical random variable. We explicitly construct states of a quantum field theory on the one-dimensional real line in which the area law is exactly satisfied.