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arXiv:hep-th/0301082 · International Journal of Modern Physics A

How Far does the Analogy Between Causal Horizon-Induced Thermalization with the Standard Heat Bath Situation go?

B. Schroer

Published May 10, 2003• 1 citation
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Abstract

After a short presentation of KMS states and modular theory as the unifying description of thermalizing systems we propose the absence of transverse vacuum fluctuations in the holographic projections as the principle reason for an area behavior (the transverse area) of localization entropy as opposed to the volume dependence of ordinary heat bath entropy. Thermalization through causal localization is not a property of QM but results from the omnipresent vacuum polarization in QFT and does not require a Gibbs type ensemble avaraging (coupling to a heat bath).

DOI
10.1142/s0217751x03015623

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After a short presentation of KMS states and modular theory as the unifying description of thermalizing systems we propose the absence of transverse vacuum fluctuations in the holographic projections as the principle reason for an area behavior (the transverse area) of localization entropy as opposed to the volume dependence of ordinary heat bath entropy. Thermalization through causal localization is not a property of QM but results from the omnipresent vacuum polarization in QFT and does not require a Gibbs type ensemble avaraging (coupling to a heat bath).
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The algebraic approach to QFT, which for several decades has enriched QFT with structural theorems, has recently shown its utility in various constru ctions of actual interest. After some introductory remarkss which illustrate this point, the mai n content of the paper addresses the issue of algebraic holography. The example of lightfront holography is presented in some detail, and its use in studying thermal aspects of localization is ex plained. The vacuum polarization at the edge of the causal horizon of a wedge-localized algebra is identified as the cause of an area proportionality of localization-entropy of localized qua ntum matter. Finally the adaptation of the method to the analogous case of black holes with bifurcated Killing horizons is sketched.
Standard entropy calculations in quantum field theory, when applied to a subsystem of definite volume, exhibit area-dependent UV divergences that make a thermodynamic interpretation troublesome. In this paper we define a renormalized entropy which is related with the Newton-Wigner position operator. Accordingly, whenever we trace over a region of space, we trace away degrees of freedom that are localized according to Newton-Wigner localization but not in the usual sense. We consider a free scalar field in d+1d+1d+1 spacetime dimensions prepared in a thermal state and we show that our entropy is free of divergences and has a perfectly sound thermodynamic behavior. In the high temperature/big volume limit our results agree with the standard QFT calculations once the divergent contributions are subtracted from the latter. In the limit of low temperature/small volume the entropy goes to zero but with a different dependence on the temperature.
We investigate the information-theoretic structure underlying causal horizons through the formalism of modular quantum channels. By analyzing the singular value decomposition of the channel induced by the partial trace over inaccessible regions, we show that the Unruh effect and the Page curve can be understood as manifestations of a universal thermal filtering mechanism governed by the modular Hamiltonian. This structure leads to a Gibbs-weighted hierarchy of information transmission modes, with entanglement entropy corresponding to spectral activation and thermal capacity. We reinterpret the first law of entanglement as a Clausius relation for modular flow and derive Einstein's equations by requiring its local validity across Rindler horizons. In the case of black hole evaporation, we find that the modular channel undergoes an informational phase transition at the Page time, marked by a shift from entanglement preservation to recoverability, quantified through fidelity and channel capacity. Based on these results, we propose the Modular Channels Flow Correspondence (MCFC): a minimal holographic principle whereby the area of a causal screen measures the storage of filtered quantum information. Our framework offers a unified operational account of holography, entropy, and curvature, grounded in the spectral dynamics of modular channels.
The algebraic approach to QFT, which for several decades has enriched QFT with structural theorems, has recently shown its utility in various constructions of actual interest. In these lecture notes I explain how AQFT (in particular the modular theory of operator algebras) implies paradigmatic conceptual and mathematical changes while fully preserving the physical principles which underly QFT. As an illustration of actual interest I use holography on null-surfaces and the ensuing area law for entropy of localized matter in the vacuum state.