2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/189533A unified S-matrix framework of quantum singular interactions is presented for the comparison of self-adjoint extensions and physical renormalization. For the long-range conformal interaction the two methods are not equivalent, with renormalization acting as selector of a preferred extension and regulator of the unbounded Hamiltonian.19 pages, including 2 figures. The title and abstract were changed to more accurately reflect the content. The text was rearranged into sections, with several equations and multiple paragraphs added for clarity; and a few typos were corrected. The central equations and concepts remain unchanged.High Energy Physics - TheoryGeneral Relativity and Quantum CosmologyMathematical PhysicsQuantum Physicstext