S-Theorem: Incompatibility of Scale Invariance and Hermiticity of Hamiltonian

In this brief note, it has been shown that scale invariance (i.e the scaling algebra [D,H] = izH) is incompatible with Hamiltonian (H) being Hermitian on a domain containing the state D|E〉, where |E〉 is the non-zero energy eigenstate and D generates scale transformation. The claim has been elucidated with the strongly coupled regime of inverse square potential and the free particle as concrete examples. The proof presented can be generalised to the statement that any operator A having a definite scaling dimension, can not be Hermitian on a domain containing D|A〉 where |A〉 is the eigenstate of operator A. As a corollary, we find that classically scale invariant system can not be made quantum without loosing either unitarity or scale invariance if we insist on having bound states with finite binding energy.