The Information in a Wave
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Communications in
Mathematical Physics
The Information in a Wave Fabio Ciolli, Roberto Longo , Giuseppe Ruzzi Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, 00133 Rome, Italy. E-mail: [email protected]; [email protected]; [email protected] Received: 6 June 2019 / Accepted: 19 August 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract: We provide the notion of entropy for a classical Klein–Gordon real wave, that we derive as particular case of a notion entropy for a vector in a Hilbert space with respect to a real linear subspace. We then consider a localised automorphism on the Rindler spacetime, in the context of a free neutral Quantum Field Theory, that is associated with a second quantised wave, and we explicitly compute its entropy S, that turns out to be given by the entropy of the associated classical wave. Here S is defined as the relative entropy between the Rindler vacuum state and the corresponding sector state (coherent state). By λ-translating the Rindler spacetime into itself along the upper null horizon, we study the behaviour of the corresponding entropy S(λ). In particular, we d2 show that the QNEC inequality in the form dλ 2 S(λ) ≥ 0 holds true for coherent states, 2
d because dλ 2 S(λ) is the integral along the space horizon of a manifestly non-negative quantity, the component of the stress-energy tensor in the null upper horizon direction.
1. Introduction Recently, the interplay between Quantum Information and Quantum Field Theory has been subject of deep analyses and new aspects and structures are appearing both in Physics and in Mathematics. The physical grounds may ultimately rely on the well know probabilistic nature of Quantum Mechanics, in particular of the wave function. Yet, the combination of General Relativity with Quantum Field Theory in the context of Black Hole Thermodynamics showed a fundamental role of Entropy related to Geometry (see [25]). Recently, the study of Entropy in relation with the Quantum Null Energy Condition has become of much interest (see [5,26] and refs therein). Supported by the ERC Advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models”, MIUR FARE R16X5RB55W QUEST-NET and GNAMPA-INdAM.
F. Ciolli, R. Longo, G. Ruzzi
Concerning Mathematics, the typical finite dimensional framework of Quantum Information is not sufficient in the QFT context, although it provides essential concepts and results that, possibly, may have an extension to the needed wider infinite dimensional context. The natural language in this context is provided by the Theory of Operator Algebras [24], in particular by the Tomita-Takesaki modular theory and by Araki’s definition of relative entropy for states of a von Neumann algebra [1]. In the Haag-Kastler approach, QFT is described by the net A of local von Neumann algebras A(O) associated to spacetime regions O [9]. In this paper, we are considering the Rindler spacetime W as embedded in the Minkowski spacetime Rd+1 , say W is the wedge region
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