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 d(2)/d lambda S-2(lambda)>= 0 holds true for coherent states, because d(2)/d lambda S-2(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.