The Information in a Wave


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 \({\lambda }\)-translating the Rindler spacetime into itself along the upper null horizon, we study the behaviour of the corresponding entropy \(S({\lambda })\). In particular, we show that the QNEC inequality in the form \(\frac{d^2}{d{\lambda }^2}S({\lambda })\ge 0\) holds true for coherent states, because \(\frac{d^2}{d{\lambda }^2}S({\lambda })\) 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.

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We acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

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Correspondence to Roberto Longo.

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Supported by the ERC Advanced Grant 669240 QUEST “Quantum Algebraic Structures and Models”, MIUR FARE R16X5RB55W QUEST-NET and GNAMPA-INdAM.

Communicated by Y. Kawahigashi

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Ciolli, F., Longo, R. & Ruzzi, G. The Information in a Wave. Commun. Math. Phys. 379, 979–1000 (2020).

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