  We compute the vacuum local modular Hamiltonian associated with a space ball
region in the free scalar massless Quantum Field Theory. We give an explicit
expression on the one particle Hilbert space in terms of the higher dimensional
Legendre differential operator. The quadratic form of the massless modular
Hamiltonian is expressed in terms of an integral of the energy density with the
parabolic distribution. We then get the formula for the local entropy of a wave
packet. This gives the vacuum relative entropy of a coherent state on the
double cone von Neumann algebras associated with the free scalar QFT. Among
other points, we provide the passivity characterisation of the modular
Hamiltonian within the standard subspace setup.
