The information in a wave

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 $\lambda$-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 $\frac{d^2}{d\lambda^2}S(\lambda)\geq 0$ 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.