Interaction of Nonrelativistic Particles with a Quantized Scalar Field

Abstract

We demonstrate the mathematical existence of a meson theory with nonrelativistic nucleons. A system of Schrödinger particles is coupled to a quantized relativistic scalar field. If a cutoff is put on the interaction, we obtain a well‐defined self‐adjoint operator. The solution of the Schrödinger equation diverges as the cutoff tends to infinity, but the divergence amounts merely to a constant infinite phase shift due to the self‐energy of the particles. In the Heisenberg picture, we obtain a solution in the limit of no cutoff. We use a canonical transformation due to Gross to separate the divergent self‐energy term. It is shown that the canonical transformation is implemented by a unitary operator, and that the transformed Hamiltonian, with an infinite constant subtracted, can be interpreted as a self‐adjoint operator.